6. Trigonometric Identities and More Equations

Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)

cos 2x = cos² x - sin² x

Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)

cos 2x = 1 - 2 sin² x

Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)

cos 2x = (cot² x - 1)/(cot² x + 1)

Match each expression in Column I with its equivalent expression in Column II.

(tan (π/3) - tan (π/4))/(1 + tan (π/3) tan (π/4))

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.

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

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. sin(60° - 45°)

In Exercises 14–19, use a sum or difference formula to find the exact value of each expression. sin 195°

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. sin 75°

Use a sum or difference formula to find the exact value of each expression. tan 5𝝅/12

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. tan ( 𝝅/3 + 𝝅/4 )

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. tan ( 5𝝅/3 ﹣ 𝝅/4)

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. sin 40° cos 20° + cos 40° sin 20°

In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression.

29. sin(5𝝅/12) cos(𝝅/4) - cos(5𝝅/12) sin(𝝅/4)

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

c. tan(α + β)

sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.

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

a. sin(α + β)

sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.