6. Trigonometric Identities and More Equations

Concept Check Suppose that sec θ = (x+4)/x.

Find an expression in x for tan θ.

Perform each indicated operation and simplify the result so that there are no quotients.

cos x/sec x + sin x/csc x

Use the given information to find each of the following.

sin 2x, given sin x = 0.6, π/2 < y < π

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

-sec² (-θ) + sin² (-θ) + cos² (-θ)

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

10. cos 67.5°

For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.

cos(-55°)

Verify that each equation is an identity.

sec⁴ x - sec² x = tan⁴ x + tan² x

If cos x = -0.750 and sin ≈ 0.6614, then tan x/2 ≈ .

Simplify each expression.

±√[(1 - cos (3θ/5))/2]

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

cos θ (cos θ - sec θ)

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working C5–C8 in the Concept and Vocabulary Check. In Exercises 9–22, express each sum or difference as a product. If possible, find this product's exact value. cos 75° ﹣ cos 15°

Verify that each equation is an identity.

sin θ/(1 - cos θ) - sin θ cos θ/( 1 + cos θ) = csc θ (1 + cos² θ)

Determine whether each statement is true or false. If false, tell why. See Example 4. cos 60° = 2 cos² 30° - 1

Verify that each equation is an identity.

cot² (x/2) = (1 + cos x)²/(sin² x)

Simplify the expression.

$\(\frac{\tan\left(-\theta\right)}{\sec\left(-\theta\right)}\)$