Trigonometric Identities

Trigonometric Identities

Section 10.4: Trigonometric Identities, from College Trigonometry: Corrected Edition by Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 license. © 2013, Carl Stitz.

770 Foundations of Trigonometry

10.4 Trigonometric Identities

In Section 10.3, we saw the utility of the Pythagorean Identities in Theorem 10.8 along with the Quotient and Reciprocal Identities in Theorem 10.6. Not only did these identities help us compute the values of the circular functions for angles, they were also useful in simplifying expressions involving the circular functions. In this section, we introduce several collections of identities which have uses in this course and beyond. Our first set of identities is the ‘Even / Odd’ identities.1

Theorem 10.12. Even / Odd Identities: For all applicable angles θ,

ˆ cos(−θ) = cos(θ)

ˆ sec(−θ) = sec(θ)

ˆ sin(−θ) = − sin(θ)

ˆ csc(−θ) = − csc(θ)

ˆ tan(−θ) = − tan(θ)

ˆ cot(−θ) = − cot(θ)

In light of the Quotient and Reciprocal Identities, Theorem 10.6, it suffices to show cos(−θ) = cos(θ) and sin(−θ) = − sin(θ). The remaining four circular functions can be expressed in terms of cos(θ) and sin(θ) so the proofs of their Even / Odd Identities are left as exercises. Consider an angle θ plotted in standard position. Let θ0 be the angle coterminal with θ with 0 ≤ θ0 < 2π. (We can construct the angle θ0 by rotating counter-clockwise from the positive x-axis to the terminal side of θ as pictured below.) Since θ and θ0 are coterminal, cos(θ) = cos(θ0) and sin(θ) = sin(θ0).

x

y

1

1

θ

θ0

x

y

1

1 θ0

−θ0

P (cos(θ0), sin(θ0))

Q(cos(−θ0), sin(−θ0))

We now consider the angles −θ and −θ0. Since θ is coterminal with θ0, there is some integer k so that θ = θ0 + 2π · k. Therefore, −θ = −θ0 − 2π · k = −θ0 + 2π · (−k). Since k is an integer, so is (−k), which means −θ is coterminal with −θ0. Hence, cos(−θ) = cos(−θ0) and sin(−θ) = sin(−θ0). Let P and Q denote the points on the terminal sides of θ0 and −θ0, respectively, which lie on the Unit Circle. By definition, the coordinates of P are (cos(θ0), sin(θ0)) and the coordinates of Q are (cos(−θ0), sin(−θ0)). Since θ0 and −θ0 sweep out congruent central sectors of the Unit Circle, it

1As mentioned at the end of Section 10.2, properties of the circular functions when thought of as functions of angles in radian measure hold equally well if we view these functions as functions of real numbers. Not surprisingly, the Even / Odd properties of the circular functions are so named because they identify cosine and secant as even functions, while the remaining four circular functions are odd. (See Section 1.6.)

10.4 Trigonometric Identities 771

follows that the points P and Q are symmetric about the x-axis. Thus, cos(−θ0) = cos(θ0) and sin(−θ0) = − sin(θ0). Since the cosines and sines of θ0 and −θ0 are the same as those for θ and −θ, respectively, we get cos(−θ) = cos(θ) and sin(−θ) = − sin(θ), as required. The Even / Odd Identities are readily demonstrated using any of the ‘common angles’ noted in Section 10.2. Their true utility, however, lies not in computation, but in simplifying expressions involving the circular functions. In fact, our next batch of identities makes heavy use of the Even / Odd Identities.

Theorem 10.13. Sum and Difference Identities for Cosine: For all angles α and β,

ˆ cos(α+ β) = cos(α) cos(β)− sin(α) sin(β)

ˆ cos(α− β) = cos(α) cos(β) + sin(α) sin(β)

We first prove the result for differences. As in the proof of the Even / Odd Identities, we can reduce the proof for general angles α and β to angles α0 and β0, coterminal with α and β, respectively, each of which measure between 0 and 2π radians. Since α and α0 are coterminal, as are β and β0, it follows that α− β is coterminal with α0 − β0. Consider the case below where α0 ≥ β0.

α0

β0

x

y

1O

P (cos(α0), sin(α0))

Q(cos(β0), sin(β0))α0 − β0

x

y

1

O

A(cos(α0 − β0), sin(α0 − β0))

B(1, 0)

α0 − β0

Since the angles POQ and AOB are congruent, the distance between P and Q is equal to the distance between A and B.2 The distance formula, Equation 1.1, yields

√ (cos(α0)− cos(β0))2 + (sin(α0)− sin(β0))2 =

√ (cos(α0 − β0)− 1)2 + (sin(α0 − β0)− 0)2

Squaring both sides, we expand the left hand side of this equation as

(cos(α0)− cos(β0))2 + (sin(α0)− sin(β0))2 = cos2(α0)− 2 cos(α0) cos(β0) + cos2(β0) + sin2(α0)− 2 sin(α0) sin(β0) + sin2(β0)

= cos2(α0) + sin 2(α0) + cos

2(β0) + sin 2(β0)

−2 cos(α0) cos(β0)− 2 sin(α0) sin(β0) 2In the picture we’ve drawn, the triangles POQ and AOB are congruent, which is even better. However, α0 − β0

could be 0 or it could be π, neither of which makes a triangle. It could also be larger than π, which makes a triangle, just not the one we’ve drawn. You should think about those three cases.

772 Foundations of Trigonometry

From the Pythagorean Identities, cos2(α0) + sin 2(α0) = 1 and cos

2(β0) + sin 2(β0) = 1, so

(cos(α0)− cos(β0))2 + (sin(α0)− sin(β0))2 = 2− 2 cos(α0) cos(β0)− 2 sin(α0) sin(β0)

Turning our attention to the right hand side of our equation, we find

(cos(α0 − β0)− 1)2 + (sin(α0 − β0)− 0)2 = cos2(α0 − β0)− 2 cos(α0 − β0) + 1 + sin2(α0 − β0) = 1 + cos2(α0 − β0) + sin2(α0 − β0)− 2 cos(α0 − β0)

Once again, we simplify cos2(α0 − β0) + sin2(α0 − β0) = 1, so that

(cos(α0 − β0)− 1)2 + (sin(α0 − β0)− 0)2 = 2− 2 cos(α0 − β0)

Putting it all together, we get 2 − 2 cos(α0) cos(β0) − 2 sin(α0) sin(β0) = 2 − 2 cos(α0 − β0), which simplifies to: cos(α0 − β0) = cos(α0) cos(β0) + sin(α0) sin(β0). Since α and α0, β and β0 and α− β and α0 − β0 are all coterminal pairs of angles, we have cos(α − β) = cos(α) cos(β) + sin(α) sin(β). For the case where α0 ≤ β0, we can apply the above argument to the angle β0 − α0 to obtain the identity cos(β0 − α0) = cos(β0) cos(α0) + sin(β0) sin(α0). Applying the Even Identity of cosine, we get cos(β0 − α0) = cos(−(α0 − β0)) = cos(α0 − β0), and we get the identity in this case, too.

To get the sum identity for cosine, we use the difference formula along with the Even/Odd Identities

cos(α+ β) = cos(α− (−β)) = cos(α) cos(−β) + sin(α) sin(−β) = cos(α) cos(β)− sin(α) sin(β)

We put these newfound identities to good use in the following example.

Example 10.4.1.

1. Find the exact value of cos (15◦).

2. Verify the identity: cos ( π 2 − θ

) = sin(θ).

Solution.

1. In order to use Theorem 10.13 to find cos (15◦), we need to write 15◦ as a sum or difference of angles whose cosines and sines we know. One way to do so is to write 15◦ = 45◦ − 30◦.

cos (15◦) = cos (45◦ − 30◦) = cos (45◦) cos (30◦) + sin (45◦) sin (30◦)

=

(√ 2

2

)(√ 3

2

) +

(√ 2

2

)( 1

2

)

=

√ 6 + √

2

4

10.4 Trigonometric Identities 773

2. In a straightforward application of Theorem 10.13, we find

cos (π

2 − θ )

= cos (π

2

) cos (θ) + sin

(π 2

) sin (θ)

= (0) (cos(θ)) + (1) (sin(θ))

= sin(θ)

The identity verified in Example 10.4.1, namely, cos ( π 2 − θ

) = sin(θ), is the first of the celebrated

‘cofunction’ identities. These identities were first hinted at in Exercise 74 in Section 10.2. From sin(θ) = cos

( π 2 − θ

) , we get:

sin (π

2 − θ )

= cos (π

2 − [π

2 − θ ])

= cos(θ),

which says, in words, that the ‘co’sine of an angle is the sine of its ‘co’mplement. Now that these identities have been established for cosine and sine, the remaining circular functions follow suit. The remaining proofs are left as exercises.

Theorem 10.14. Cofunction Identities: For all applicable angles θ,

ˆ cos (π

2 − θ )

= sin(θ)

ˆ sin (π

2 − θ )

= cos(θ)

ˆ sec (π

2 − θ )

= csc(θ)

ˆ csc (π

2 − θ )

= sec(θ)

ˆ tan (π

2 − θ )

= cot(θ)

ˆ cot (π

2 − θ )

= tan(θ)

With the Cofunction Identities in place, we are now in the position to derive the sum and difference formulas for sine. To derive the sum formula for sine, we convert to cosines using a cofunction identity, then expand using the difference formula for cosine

sin(α+ β) = cos (π

2 − (α+ β)

) = cos

([π 2 − α

] − β

) = cos

(π 2 − α

) cos(β) + sin

(π 2 − α

) sin(β)

= sin(α) cos(β) + cos(α) sin(β)

We can derive the difference formula for sine by rewriting sin(α − β) as sin(α + (−β)) and using the sum formula and the Even / Odd Identities. Again, we leave the details to the reader.

Theorem 10.15. Sum and Difference Identities for Sine: For all angles α and β,

ˆ sin(α+ β) = sin(α) cos(β) + cos(α) sin(β)

ˆ sin(α− β) = sin(α) cos(β)− cos(α) sin(β)

774 Foundations of Trigonometry

Example 10.4.2.

1. Find the exact value of sin (

19π 12

) 2. If α is a Quadrant II angle with sin(α) = 513 , and β is a Quadrant III angle with tan(β) = 2,

find sin(α− β).

3. Derive a formula for tan(α+ β) in terms of tan(α) and tan(β).

Solution.

1. As in Example 10.4.1, we need to write the angle 19π12 as a sum or difference of common angles. The denominator of 12 suggests a combination of angles with denominators 3 and 4. One such combination is 19π12 =

4π 3 +

π 4 . Applying Theorem 10.15, we get

sin

( 19π

12

) = sin

( 4π

3 + π

4

) = sin

( 4π

3

) cos (π

4

) + cos

( 4π

3

) sin (π

4

) =

( − √

3

2

)(√ 2

2

) +

( −1

2

)(√ 2

2

)

= − √

6− √

2

4

2. In order to find sin(α − β) using Theorem 10.15, we need to find cos(α) and both cos(β) and sin(β). To find cos(α), we use the Pythagorean Identity cos2(α) + sin2(α) = 1. Since

sin(α) = 513 , we have cos 2(α) +

( 5 13

)2 = 1, or cos(α) = ±1213 . Since α is a Quadrant II angle,

cos(α) = −1213 . We now set about finding cos(β) and sin(β). We have several ways to proceed, but the Pythagorean Identity 1 + tan2(β) = sec2(β) is a quick way to get sec(β), and hence, cos(β). With tan(β) = 2, we get 1 + 22 = sec2(β) so that sec(β) = ±

√ 5. Since β is a

Quadrant III angle, we choose sec(β) = − √

5 so cos(β) = 1sec(β) = 1 − √

5 = −

√ 5

5 . We now need

to determine sin(β). We could use The Pythagorean Identity cos2(β) + sin2(β) = 1, but we

opt instead to use a quotient identity. From tan(β) = sin(β)cos(β) , we have sin(β) = tan(β) cos(β)

so we get sin(β) = (2) ( − √

5 5

) = −2

√ 5

5 . We now have all the pieces needed to find sin(α−β):

sin(α− β) = sin(α) cos(β)− cos(α) sin(β)

=

( 5

13

)( − √

5

5

) − ( −12

13

)( −2 √

5

5

) = −29

√ 5

65

10.4 Trigonometric Identities 775

3. We can start expanding tan(α+ β) using a quotient identity and our sum formulas

tan(α+ β) = sin(α+ β)

cos(α+ β)

= sin(α) cos(β) + cos(α) sin(β)

cos(α) cos(β)− sin(α) sin(β)

Since tan(α) = sin(α)cos(α) and tan(β) = sin(β) cos(β) , it looks as though if we divide both numerator and

denominator by cos(α) cos(β) we will have what we want

tan(α+ β) = sin(α) cos(β) + cos(α) sin(β)

cos(α) cos(β)− sin(α) sin(β) ·

1

cos(α) cos(β) 1

cos(α) cos(β)

=

sin(α) cos(β)

cos(α) cos(β) +

cos(α) sin(β)

cos(α) cos(β)

cos(α) cos(β)

cos(α) cos(β) − sin(α) sin(β)

cos(α) cos(β)

=

sin(α)��� �cos(β)

cos(α)��� �cos(β)

+ ���

�cos(α) sin(β)

��� �cos(α) cos(β)

�� ��cos(α)���

�cos(β)

�� ��cos(α)���

�cos(β) − sin(α) sin(β)

cos(α) cos(β)

= tan(α) + tan(β)

1− tan(α) tan(β)

Naturally, this formula is limited to those cases where all of the tangents are defined.

The formula developed in Exercise 10.4.2 for tan(α+β) can be used to find a formula for tan(α−β) by rewriting the difference as a sum, tan(α+(−β)), and the reader is encouraged to fill in the details. Below we summarize all of the sum and difference formulas for cosine, sine and tangent.

Theorem 10.16. Sum and Difference Identities: For all applicable angles α and β,

ˆ cos(α± β) = cos(α) cos(β)∓ sin(α) sin(β)

ˆ sin(α± β) = sin(α) cos(β)± cos(α) sin(β)

ˆ tan(α± β) = tan(α)± tan(β) 1∓ tan(α) tan(β)

In the statement of Theorem 10.16, we have combined the cases for the sum ‘+’ and difference ‘−’ of angles into one formula. The convention here is that if you want the formula for the sum ‘+’ of

776 Foundations of Trigonometry

two angles, you use the top sign in the formula; for the difference, ‘−’, use the bottom sign. For example,

tan(α− β) = tan(α)− tan(β) 1 + tan(α) tan(β)

If we specialize the sum formulas in Theorem 10.16 to the case when α = β, we obtain the following ‘Double Angle’ Identities.

Theorem 10.17. Double Angle Identities: For all applicable angles θ,

ˆ cos(2θ) =

 cos2(θ)− sin2(θ) 2 cos2(θ)− 1 1− 2 sin2(θ)

ˆ sin(2θ) = 2 sin(θ) cos(θ)

ˆ tan(2θ) = 2 tan(θ)

1− tan2(θ)

The three different forms for cos(2θ) can be explained by our ability to ‘exchange’ squares of cosine and sine via the Pythagorean Identity cos2(θ) + sin2(θ) = 1 and we leave the details to the reader. It is interesting to note that to determine the value of cos(2θ), only one piece of information is required: either cos(θ) or sin(θ). To determine sin(2θ), however, it appears that we must know both sin(θ) and cos(θ). In the next example, we show how we can find sin(2θ) knowing just one piece of information, namely tan(θ).

Example 10.4.3.

1. Suppose P (−3, 4) lies on the terminal side of θ when θ is plotted in standard position. Find cos(2θ) and sin(2θ) and determine the quadrant in which the terminal side of the angle 2θ lies when it is plotted in standard position.

2. If sin(θ) = x for −π2 ≤ θ ≤ π 2 , find an expression for sin(2θ) in terms of x.

3. Verify the identity: sin(2θ) = 2 tan(θ)

1 + tan2(θ) .

4. Express cos(3θ) as a polynomial in terms of cos(θ).

Solution.

1. Using Theorem 10.3 from Section 10.2 with x = −3 and y = 4, we find r = √ x2 + y2 = 5.

Hence, cos(θ) = −35 and sin(θ) = 4 5 . Applying Theorem 10.17, we get cos(2θ) = cos

2(θ) − sin2(θ) =

( −35 )2− (45)2 = − 725 , and sin(2θ) = 2 sin(θ) cos(θ) = 2 (45) (−35) = −2425 . Since both

cosine and sine of 2θ are negative, the terminal side of 2θ, when plotted in standard position, lies in Quadrant III.

10.4 Trigonometric Identities 777

2. If your first reaction to ‘sin(θ) = x’ is ‘No it’s not, cos(θ) = x!’ then you have indeed learned something, and we take comfort in that. However, context is everything. Here, ‘x’ is just a variable – it does not necessarily represent the x-coordinate of the point on The Unit Circle which lies on the terminal side of θ, assuming θ is drawn in standard position. Here, x represents the quantity sin(θ), and what we wish to know is how to express sin(2θ) in terms of x. We will see more of this kind of thing in Section 10.6, and, as usual, this is something we need for Calculus. Since sin(2θ) = 2 sin(θ) cos(θ), we need to write cos(θ) in terms of x to finish the problem. We substitute x = sin(θ) into the Pythagorean Identity, cos2(θ) + sin2(θ) = 1, to get cos2(θ) + x2 = 1, or cos(θ) = ±

√ 1− x2. Since −π2 ≤ θ ≤

π 2 , cos(θ) ≥ 0, and thus

cos(θ) = √

1− x2. Our final answer is sin(2θ) = 2 sin(θ) cos(θ) = 2x √

1− x2.

3. We start with the right hand side of the identity and note that 1 + tan2(θ) = sec2(θ). From this point, we use the Reciprocal and Quotient Identities to rewrite tan(θ) and sec(θ) in terms of cos(θ) and sin(θ):

2 tan(θ)

1 + tan2(θ) =

2 tan(θ)

sec2(θ) =

2

( sin(θ)

cos(θ)

) 1

cos2(θ)

= 2

( sin(θ)

cos(θ)

) cos2(θ)

= 2

( sin(θ)

��� �cos(θ)

) ���

�cos(θ) cos(θ) = 2 sin(θ) cos(θ) = sin(2θ)

4. In Theorem 10.17, one of the formulas for cos(2θ), namely cos(2θ) = 2 cos2(θ)− 1, expresses cos(2θ) as a polynomial in terms of cos(θ). We are now asked to find such an identity for cos(3θ). Using the sum formula for cosine, we begin with

cos(3θ) = cos(2θ + θ)

= cos(2θ) cos(θ)− sin(2θ) sin(θ)

Our ultimate goal is to express the right hand side in terms of cos(θ) only. We substitute cos(2θ) = 2 cos2(θ)− 1 and sin(2θ) = 2 sin(θ) cos(θ) which yields

cos(3θ) = cos(2θ) cos(θ)− sin(2θ) sin(θ) =

( 2 cos2(θ)− 1

) cos(θ)− (2 sin(θ) cos(θ)) sin(θ)

= 2 cos3(θ)− cos(θ)− 2 sin2(θ) cos(θ)

Finally, we exchange sin2(θ) for 1− cos2(θ) courtesy of the Pythagorean Identity, and get

cos(3θ) = 2 cos3(θ)− cos(θ)− 2 sin2(θ) cos(θ) = 2 cos3(θ)− cos(θ)− 2

( 1− cos2(θ)

) cos(θ)

= 2 cos3(θ)− cos(θ)− 2 cos(θ) + 2 cos3(θ) = 4 cos3(θ)− 3 cos(θ)

and we are done.

778 Foundations of Trigonometry

In the last problem in Example 10.4.3, we saw how we could rewrite cos(3θ) as sums of powers of cos(θ). In Calculus, we have occasion to do the reverse; that is, reduce the power of cosine and sine. Solving the identity cos(2θ) = 2 cos2(θ)−1 for cos2(θ) and the identity cos(2θ) = 1−2 sin2(θ) for sin2(θ) results in the aptly-named ‘Power Reduction’ formulas below.

Theorem 10.18. Power Reduction Formulas: For all angles θ,

ˆ cos2(θ) = 1 + cos(2θ)

2

ˆ sin2(θ) = 1− cos(2θ)

2

Example 10.4.4. Rewrite sin2(θ) cos2(θ) as a sum and difference of cosines to the first power.

Solution. We begin with a straightforward application of Theorem 10.18

sin2(θ) cos2(θ) =

( 1− cos(2θ)

2

)( 1 + cos(2θ)

2

) =

1

4

( 1− cos2(2θ)

) =

1

4 − 1

4 cos2(2θ)

Next, we apply the power reduction formula to cos2(2θ) to finish the reduction

sin2(θ) cos2(θ) = 1

4 − 1

4 cos2(2θ)

= 1

4 − 1

4

( 1 + cos(2(2θ))

2

) =

1

4 − 1

8 − 1

8 cos(4θ)

= 1

8 − 1

8 cos(4θ)

Another application of the Power Reduction Formulas is the Half Angle Formulas. To start, we apply the Power Reduction Formula to cos2

( θ 2

) cos2

( θ

2

) =

1 + cos ( 2 ( θ 2

)) 2

= 1 + cos(θ)

2 .

We can obtain a formula for cos ( θ 2

) by extracting square roots. In a similar fashion, we may obtain

a half angle formula for sine, and by using a quotient formula, obtain a half angle formula for tangent. We summarize these formulas below.

10.4 Trigonometric Identities 779

Theorem 10.19. Half Angle Formulas: For all applicable angles θ,

ˆ cos

( θ

2

) = ±

√ 1 + cos(θ)

2

ˆ sin

( θ

2

) = ±

√ 1− cos(θ)

2

ˆ tan

( θ

2

) = ±

√ 1− cos(θ) 1 + cos(θ)

where the choice of ± depends on the quadrant in which the terminal side of θ 2

lies.

Example 10.4.5.

1. Use a half angle formula to find the exact value of cos (15◦).

2. Suppose −π ≤ θ ≤ 0 with cos(θ) = −35 . Find sin ( θ 2

) .

3. Use the identity given in number 3 of Example 10.4.3 to derive the identity

tan

( θ

2

) =

sin(θ)

1 + cos(θ)

Solution.

1. To use the half angle formula, we note that 15◦ = 30 ◦

2 and since 15 ◦ is a Quadrant I angle,

its cosine is positive. Thus we have

cos (15◦) = +

√ 1 + cos (30◦)

2 =

√ 1 +

√ 3

2

2

=

√ 1 +

√ 3

2

2 · 2

2 =

√ 2 + √

3

4 =

√ 2 + √

3

2

Back in Example 10.4.1, we found cos (15◦) by using the difference formula for cosine. In that

case, we determined cos (15◦) = √

6+ √

2 4 . The reader is encouraged to prove that these two

expressions are equal.

2. If −π ≤ θ ≤ 0, then −π2 ≤ θ 2 ≤ 0, which means sin

( θ 2

) < 0. Theorem 10.19 gives

sin

( θ

2

) = −

√ 1− cos (θ)

2 = −

√ 1−

( −35 )

2

= − √

1 + 35 2 · 5

5 = −

√ 8

10 = −2

√ 5

5

780 Foundations of Trigonometry

3. Instead of our usual approach to verifying identities, namely starting with one side of the equation and trying to transform it into the other, we will start with the identity we proved in number 3 of Example 10.4.3 and manipulate it into the identity we are asked to prove. The identity we are asked to start with is sin(2θ) = 2 tan(θ)

1+tan2(θ) . If we are to use this to derive an

identity for tan ( θ 2

) , it seems reasonable to proceed by replacing each occurrence of θ with θ2

sin ( 2 ( θ 2

)) =

2 tan ( θ 2

) 1 + tan2

( θ 2

) sin(θ) =

2 tan ( θ 2

) 1 + tan2

( θ 2

) We now have the sin(θ) we need, but we somehow need to get a factor of 1 + cos(θ) involved. To get cosines involved, recall that 1 + tan2

( θ 2

) = sec2

( θ 2

) . We continue to manipulate our

given identity by converting secants to cosines and using a power reduction formula

sin(θ) = 2 tan

( θ 2

) 1 + tan2

( θ 2

) sin(θ) =

2 tan ( θ 2

) sec2

( θ 2

) sin(θ) = 2 tan

( θ 2

) cos2

( θ 2

) sin(θ) = 2 tan

( θ 2

)(1 + cos (2 ( θ2)) 2

) sin(θ) = tan

( θ 2

) (1 + cos(θ))

tan

( θ

2

) =

sin(θ)

1 + cos(θ)

Our next batch of identities, the Product to Sum Formulas,3 are easily verified by expanding each of the right hand sides in accordance with Theorem 10.16 and as you should expect by now we leave the details as exercises. They are of particular use in Calculus, and we list them here for reference.

Theorem 10.20. Product to Sum Formulas: For all angles α and β,

ˆ cos(α) cos(β) = 12 [cos(α− β) + cos(α+ β)]

ˆ sin(α) sin(β) = 12 [cos(α− β)− cos(α+ β)]

ˆ sin(α) cos(β) = 12 [sin(α− β) + sin(α+ β)]

3These are also known as the Prosthaphaeresis Formulas and have a rich history. The authors recommend that you conduct some research on them as your schedule allows.

10.4 Trigonometric Identities 781

Related to the Product to Sum Formulas are the Sum to Product Formulas, which we will have need of in Section 10.7. These are easily verified using the Product to Sum Formulas, and as such, their proofs are left as exercises.

Theorem 10.21. Sum to Product Formulas: For all angles α and β,

ˆ cos(α) + cos(β) = 2 cos

( α+ β

2

) cos

( α− β

2

)

ˆ cos(α)− cos(β) = −2 sin ( α+ β

2

) sin

( α− β

2

)

ˆ sin(α)± sin(β) = 2 sin ( α± β

2

) cos

( α∓ β

2

)

Example 10.4.6.

1. Write cos(2θ) cos(6θ) as a sum.

2. Write sin(θ)− sin(3θ) as a product.

Solution.

1. Identifying α = 2θ and β = 6θ, we find

cos(2θ) cos(6θ) = 12 [cos(2θ − 6θ) + cos(2θ + 6θ)] = 12 cos(−4θ) +

1 2 cos(8θ)

= 12 cos(4θ) + 1 2 cos(8θ),

where the last equality is courtesy of the even identity for cosine, cos(−4θ) = cos(4θ).

2. Identifying α = θ and β = 3θ yields

sin(θ)− sin(3θ) = 2 sin ( θ − 3θ

2

) cos

( θ + 3θ

2

) = 2 sin (−θ) cos (2θ) = −2 sin (θ) cos (2θ) ,

where the last equality is courtesy of the odd identity for sine, sin(−θ) = − sin(θ).

The reader is reminded that all of the identities presented in this section which regard the circular functions as functions of angles (in radian measure) apply equally well to the circular (trigonometric) functions regarded as functions of real numbers. In Exercises 38 – 43 in Section 10.5, we see how some of these identities manifest themselves geometrically as we study the graphs of the these functions. In the upcoming Exercises, however, you need to do all of your work analytically without graphs.

782 Foundations of Trigonometry

10.4.1 Exercises

In Exercises 1 – 6, use the Even / Odd Identities to verify the identity. Assume all quantities are defined.

1. sin(3π − 2θ) = − sin(2θ − 3π) 2. cos ( −π

4 − 5t

) = cos

( 5t+

π

4

) 3. tan(−t2 + 1) = − tan(t2 − 1) 4. csc(−θ − 5) = − csc(θ + 5)

5. sec(−6t) = sec(6t) 6. cot(9− 7θ) = − cot(7θ − 9)

In Exercises 7 – 21, use the Sum and Difference Identities to find the exact value. You may have need of the Quotient, Reciprocal or Even / Odd Identities as well.

7. cos(75◦) 8. sec(165◦) 9. sin(105◦)

10. csc(195◦) 11. cot(255◦) 12. tan(375◦)

13. cos

( 13π

12

) 14. sin

( 11π

12

) 15. tan

( 13π

12

)

16. cos

( 7π

12

) 17. tan

( 17π

12

) 18. sin

( π 12

) 19. cot

( 11π

12

) 20. csc

( 5π

12

) 21. sec

( − π

12

) 22. If α is a Quadrant IV angle with cos(α) =

√ 5

5 , and sin(β) =

√ 10

10 , where

π

2 < β < π, find

(a) cos(α+ β) (b) sin(α+ β) (c) tan(α+ β)

(d) cos(α− β) (e) sin(α− β) (f) tan(α− β)

23. If csc(α) = 3, where 0 < α < π

2 , and β is a Quadrant II angle with tan(β) = −7, find

(a) cos(α+ β) (b) sin(α+ β) (c) tan(α+ β)

(d) cos(α− β) (e) sin(α− β) (f) tan(α− β)

24. If sin(α) = 3

5 , where 0 < α <

π

2 , and cos(β) =

12

13 where

2 < β < 2π, find

(a) sin(α+ β) (b) cos(α− β) (c) tan(α− β)

10.4 Trigonometric Identities 783

25. If sec(α) = −5 3

, where π

2 < α < π, and tan(β) =

24

7 , where π < β <

2 , find

(a) csc(α− β) (b) sec(α+ β) (c) cot(α+ β)

In Exercises 26 – 38, verify the identity.

26. cos(θ − π) = − cos(θ) 27. sin(π − θ) = sin(θ)

28. tan ( θ +

π

2

) = − cot(θ) 29. sin(α+ β) + sin(α− β) = 2 sin(α) cos(β)

30. sin(α+ β)− sin(α− β) = 2 cos(α) sin(β) 31. cos(α+ β) + cos(α− β) = 2 cos(α) cos(β)

32. cos(α+β)− cos(α−β) = −2 sin(α) sin(β) 33. sin(α+ β) sin(α− β)

= 1 + cot(α) tan(β)

1− cot(α) tan(β)

34. cos(α+ β)

cos(α− β) =

1− tan(α) tan(β) 1 + tan(α) tan(β)

35. tan(α+ β)

tan(α− β) =

sin(α) cos(α) + sin(β) cos(β)

sin(α) cos(α)− sin(β) cos(β)

36. sin(t+ h)− sin(t)

h = cos(t)

( sin(h)

h

) + sin(t)

( cos(h)− 1

h

)

37. cos(t+ h)− cos(t)

h = cos(t)

( cos(h)− 1

h

) − sin(t)

( sin(h)

h

)

38. tan(t+ h)− tan(t)

h =

( tan(h)

h

)( sec2(t)

1− tan(t) tan(h)

) In Exercises 39 – 48, use the Half Angle Formulas to find the exact value. You may have need of the Quotient, Reciprocal or Even / Odd Identities as well.

39. cos(75◦) (compare with Exercise 7) 40. sin(105◦) (compare with Exercise 9)

41. cos(67.5◦) 42. sin(157.5◦)

43. tan(112.5◦) 44. cos

( 7π

12

) (compare with Exercise 16)

45. sin ( π

12

) (compare with Exercise 18) 46. cos

(π 8

) 47. sin

( 5π

8

) 48. tan

( 7π

8

)

784 Foundations of Trigonometry

In Exercises 49 – 58, use the given information about θ to find the exact values of

ˆ sin(2θ)

ˆ sin

( θ

2

) ˆ cos(2θ) ˆ cos

( θ

2

) ˆ tan(2θ) ˆ tan

( θ

2

) 49. sin(θ) = − 7

25 where

2 < θ < 2π 50. cos(θ) =

28

53 where 0 < θ <

π

2

51. tan(θ) = 12

5 where π < θ <

2 52. csc(θ) = 4 where

π

2 < θ < π

53. cos(θ) = 3

5 where 0 < θ <

π

2 54. sin(θ) = −4

5 where π < θ <

2

55. cos(θ) = 12

13 where

2 < θ < 2π 56. sin(θ) =

5

13 where

π

2 < θ < π

57. sec(θ) = √

5 where 3π

2 < θ < 2π 58. tan(θ) = −2 where

π

2 < θ < π

In Exercises 59 – 73, verify the identity. Assume all quantities are defined.

59. (cos(θ) + sin(θ))2 = 1 + sin(2θ) 60. (cos(θ)− sin(θ))2 = 1− sin(2θ)

61. tan(2θ) = 1

1− tan(θ) − 1

1 + tan(θ) 62. csc(2θ) =

cot(θ) + tan(θ)

2

63. 8 sin4(θ) = cos(4θ)− 4 cos(2θ) + 3 64. 8 cos4(θ) = cos(4θ) + 4 cos(2θ) + 3

65. sin(3θ) = 3 sin(θ)− 4 sin3(θ) 66. sin(4θ) = 4 sin(θ) cos3(θ)− 4 sin3(θ) cos(θ)

67. 32 sin2(θ) cos4(θ) = 2 + cos(2θ)− 2 cos(4θ)− cos(6θ)

68. 32 sin4(θ) cos2(θ) = 2− cos(2θ)− 2 cos(4θ) + cos(6θ)

69. cos(4θ) = 8 cos4(θ)− 8 cos2(θ) + 1

70. cos(8θ) = 128 cos8(θ)−256 cos6(θ)+160 cos4(θ)−32 cos2(θ)+1 (HINT: Use the result to 69.)

71. sec(2θ) = cos(θ)

cos(θ) + sin(θ) +

sin(θ)

cos(θ)− sin(θ)

72. 1

cos(θ)− sin(θ) +

1

cos(θ) + sin(θ) =

2 cos(θ)

cos(2θ)

73. 1

cos(θ)− sin(θ) − 1

cos(θ) + sin(θ) =

2 sin(θ)

cos(2θ)

10.4 Trigonometric Identities 785

In Exercises 74 – 79, write the given product as a sum. You may need to use an Even/Odd Identity.

74. cos(3θ) cos(5θ) 75. sin(2θ) sin(7θ) 76. sin(9θ) cos(θ)

77. cos(2θ) cos(6θ) 78. sin(3θ) sin(2θ) 79. cos(θ) sin(3θ)

In Exercises 80 – 85, write the given sum as a product. You may need to use an Even/Odd or Cofunction Identity.

80. cos(3θ) + cos(5θ) 81. sin(2θ)− sin(7θ) 82. cos(5θ)− cos(6θ)

83. sin(9θ)− sin(−θ) 84. sin(θ) + cos(θ) 85. cos(θ)− sin(θ)

86. Suppose θ is a Quadrant I angle with sin(θ) = x. Verify the following formulas

(a) cos(θ) = √

1− x2 (b) sin(2θ) = 2x √

1− x2 (c) cos(2θ) = 1− 2×2

87. Discuss with your classmates how each of the formulas, if any, in Exercise 86 change if we change assume θ is a Quadrant II, III, or IV angle.

88. Suppose θ is a Quadrant I angle with tan(θ) = x. Verify the following formulas

(a) cos(θ) = 1√

x2 + 1 (b) sin(θ) =

x√ x2 + 1

(c) sin(2θ) = 2x

x2 + 1 (d) cos(2θ) = 1− x2

x2 + 1

89. Discuss with your classmates how each of the formulas, if any, in Exercise 88 change if we change assume θ is a Quadrant II, III, or IV angle.

90. If sin(θ) = x

2 for −π

2 < θ <

π

2 , find an expression for cos(2θ) in terms of x.

91. If tan(θ) = x

7 for −π

2 < θ <

π

2 , find an expression for sin(2θ) in terms of x.

92. If sec(θ) = x

4 for 0 < θ <

π

2 , find an expression for ln | sec(θ) + tan(θ)| in terms of x.

93. Show that cos2(θ)− sin2(θ) = 2 cos2(θ)− 1 = 1− 2 sin2(θ) for all θ.

94. Let θ be a Quadrant III angle with cos(θ) = −1 5

. Show that this is not enough information to

determine the sign of sin

( θ

2

) by first assuming 3π < θ <

2 and then assuming π < θ <

2

and computing sin

( θ

2

) in both cases.

786 Foundations of Trigonometry

95. Without using your calculator, show that

√ 2 + √

3

2 =

√ 6 + √

2

4

96. In part 4 of Example 10.4.3, we wrote cos(3θ) as a polynomial in terms of cos(θ). In Exercise 69, we had you verify an identity which expresses cos(4θ) as a polynomial in terms of cos(θ). Can you find a polynomial in terms of cos(θ) for cos(5θ)? cos(6θ)? Can you find a pattern so that cos(nθ) could be written as a polynomial in cosine for any natural number n?

97. In Exercise 65, we has you verify an identity which expresses sin(3θ) as a polynomial in terms of sin(θ). Can you do the same for sin(5θ)? What about for sin(4θ)? If not, what goes wrong?

98. Verify the Even / Odd Identities for tangent, secant, cosecant and cotangent.

99. Verify the Cofunction Identities for tangent, secant, cosecant and cotangent.

100. Verify the Difference Identities for sine and tangent.

101. Verify the Product to Sum Identities.

102. Verify the Sum to Product Identities.

10.4 Trigonometric Identities 787

10.4.2 Answers

7. cos(75◦) =

√ 6− √

2

4 8. sec(165◦) = − 4√

2 + √

6 = √

2− √

6

9. sin(105◦) =

√ 6 + √

2

4 10. csc(195◦) =

4√ 2− √

6 = −(

√ 2 + √

6)

11. cot(255◦) =

√ 3− 1√ 3 + 1

= 2− √

3 12. tan(375◦) = 3− √

3

3 + √

3 = 2−

√ 3

13. cos

( 13π

12

) = − √

6 + √

2

4 14. sin

( 11π

12

) =

√ 6− √

2

4

15. tan

( 13π

12

) =

3− √

3

3 + √

3 = 2−

√ 3 16. cos

( 7π

12

) =

√ 2− √

6

4

17. tan

( 17π

12

) = 2 +

√ 3 18. sin

( π 12

) =

√ 6− √

2

4

19. cot

( 11π

12

) = −(2 +

√ 3) 20. csc

( 5π

12

) = √

6− √

2

21. sec ( − π

12

) = √

6− √

2

22. (a) cos(α+ β) = − √

2

10 (b) sin(α+ β) =

7 √

2

10

(c) tan(α+ β) = −7 (d) cos(α− β) = − √

2

2

(e) sin(α− β) = √

2

2 (f) tan(α− β) = −1

23. (a) cos(α+ β) = −4 + 7 √

2

30 (b) sin(α+ β) =

28− √

2

30

(c) tan(α+ β) = −28 +

√ 2

4 + 7 √

2 =

63− 100 √

2

41 (d) cos(α− β) = −4 + 7

√ 2

30

(e) sin(α− β) = −28 + √

2

30 (f) tan(α− β) = 28 +

√ 2

4− 7 √

2 = −63 + 100

√ 2

41

24. (a) sin(α+ β) = 16

65 (b) cos(α− β) = 33

65 (c) tan(α− β) = 56

33

788 Foundations of Trigonometry

25. (a) csc(α− β) = −5 4

(b) sec(α+ β) = 125

117 (c) cot(α+ β) =

117

44

39. cos(75◦) =

√ 2− √

3

2 40. sin(105◦) =

√ 2 + √

3

2

41. cos(67.5◦) =

√ 2− √

2

2 42. sin(157.5◦) =

√ 2− √

2

2

43. tan(112.5◦) = −

√ 2 + √

2

2− √

2 = −1−

√ 2 44. cos

( 7π

12

) = −

√ 2− √

3

2

45. sin ( π

12

) =

√ 2− √

3

2 46. cos

(π 8

) =

√ 2 + √

2

2

47. sin

( 5π

8

) =

√ 2 + √

2

2 48. tan

( 7π

8

) = −

√ 2− √

2

2 + √

2 = 1−

√ 2

49. ˆ sin(2θ) = −336 625

ˆ sin ( θ 2

) =

√ 2

10

ˆ cos(2θ) = 527

625

ˆ cos ( θ 2

) = −7

√ 2

10

ˆ tan(2θ) = −336 527

ˆ tan ( θ 2

) = −1

7

50. ˆ sin(2θ) = 2520

2809

ˆ sin ( θ 2

) =

5 √

106

106

ˆ cos(2θ) = −1241 2809

ˆ cos ( θ 2

) =

9 √

106

106

ˆ tan(2θ) = −2520 1241

ˆ tan ( θ 2

) =

5

9

51. ˆ sin(2θ) = 120

169

ˆ sin ( θ 2

) =

3 √

13

13

ˆ cos(2θ) = −119 169

ˆ cos ( θ 2

) = −2

√ 13

13

ˆ tan(2θ) = −120 119

ˆ tan ( θ 2

) = −3

2

52. ˆ sin(2θ) = − √

15

8

ˆ sin ( θ 2

) =

√ 8 + 2

√ 15

4

ˆ cos(2θ) = 7

8

ˆ cos ( θ 2

) =

√ 8− 2

√ 15

4

ˆ tan(2θ) = − √

15

7

ˆ tan ( θ 2

) =

√ 8 + 2

√ 15

8− 2 √

15

tan ( θ 2

) = 4 +

√ 15

53. ˆ sin(2θ) = 24

25

ˆ sin ( θ 2

) =

√ 5

5

ˆ cos(2θ) = − 7 25

ˆ cos ( θ 2

) =

2 √

5

5

ˆ tan(2θ) = −24 7

ˆ tan ( θ 2

) =

1

2

10.4 Trigonometric Identities 789

54. ˆ sin(2θ) = 24

25

ˆ sin ( θ 2

) =

2 √

5

5

ˆ cos(2θ) = − 7 25

ˆ cos ( θ 2

) = − √

5

5

ˆ tan(2θ) = −24 7

ˆ tan ( θ 2

) = −2

55. ˆ sin(2θ) = −120 169

ˆ sin ( θ 2

) =

√ 26

26

ˆ cos(2θ) = 119

169

ˆ cos ( θ 2

) = −5

√ 26

26

ˆ tan(2θ) = −120 119

ˆ tan ( θ 2

) = −1

5

56. ˆ sin(2θ) = −120 169

ˆ sin ( θ 2

) =

5 √

26

26

ˆ cos(2θ) = 119

169

ˆ cos ( θ 2

) =

√ 26

26

ˆ tan(2θ) = −120 119

ˆ tan ( θ 2

) = 5

57. ˆ sin(2θ) = −4 5

ˆ sin ( θ 2

) =

√ 50− 10

√ 5

10

ˆ cos(2θ) = −3 5

ˆ cos ( θ 2

) = −

√ 50 + 10

√ 5

10

ˆ tan(2θ) = 4

3

ˆ tan ( θ 2

) = −

√ 5− √

5

5 + √

5

tan ( θ 2

) =

5− 5 √

5

10

58. ˆ sin(2θ) = −4 5

ˆ sin ( θ 2

) =

√ 50 + 10

√ 5

10

ˆ cos(2θ) = −3 5

ˆ cos ( θ 2

) =

√ 50− 10

√ 5

10

ˆ tan(2θ) = 4

3

ˆ tan ( θ 2

) =

√ 5 + √

5

5− √

5

tan ( θ 2

) =

5 + 5 √

5

10

74. cos(2θ) + cos(8θ)

2 75.

cos(5θ)− cos(9θ) 2

76. sin(8θ) + sin(10θ)

2

77. cos(4θ) + cos(8θ)

2 78.

cos(θ)− cos(5θ) 2

79. sin(2θ) + sin(4θ)

2

80. 2 cos(4θ) cos(θ) 81. −2 cos (

9

2 θ

) sin

( 5

2 θ

) 82. 2 sin

( 11

2 θ

) sin

( 1

2 θ

) 83. 2 cos(4θ) sin(5θ) 84.

√ 2 cos

( θ − π

4

) 85. −

√ 2 sin

( θ − π

4

) 90. 1− x

2

2 91.

14x

x2 + 49 92. ln |x+

√ x2 + 16| − ln(4)


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