6. Trigonometric Identities and More Equations

Find the exact value of each expression. See Example 1.

[tan 80° - tan(-55°)]/[ 1 + tan 80° tan(-55°)]

Verify that each equation is an identity.

(tan(α + β) - tan β)/(1 + tan(α + β) tan β) = tan α

Each expression is the right side of the formula for cos (α - β) with particular values for α and β. Find the exact value of the expression.

$\cos \frac{5\pi }{12}\cos \frac{\pi }{12}+\sin \frac{5\pi }{12}\sin \frac{\pi }{12}$

Write each function as an expression involving functions of θ or x alone. See Example 2.

sin (3π/4 - x)

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. cos(135° + 30°)

Use identities to fill in each blank with the appropriate trigonometric function name.

____ 72° = cot 18°

Find one value of θ or x that satisfies each of the following.

cos x = sin (π/12)

In Exercises 35–38, find the exact value of the following under the given conditions:

b. cos(α﹣β)

sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.

Use the given information to find tan(s + t). See Example 3.

sin s = 3/5 and sin t = -12/13, s in quadrant I and t in quadrant III

Find one value of θ or x that satisfies each of the following.

cot(θ - 10°) = tan(2θ - 20°)

In Exercises 57–64, find the exact value of the following under the given conditions: c. tan (α + β), sin α = 5/6 , 𝝅/2 < α < 𝝅 , and tan β = 3/7 , 𝝅 < β < 3𝝅/2 .

Find the exact value of each expression.

sin (- 5π/12)

Find cos(s + t) and cos(s - t).

cos s = - 8/17 and cos t = - 3/5, s and t in quadrant III

Find the exact value of each expression.

sin (π/12)

Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference.

$\cos \frac{3x}{2}\sin \frac{x}{2}$