6. Trigonometric Identities and More Equations

Simplify the expression.

$\(\left\)(\(\frac{\tan^2\theta}{\sin^2\theta}\)-1\(\right\))\(\csc\)^2\(\left\)(\(\theta\[\right\))\(\cos\)^2\(\left\)(-\(\theta\]\right\))$

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.

-sin 35°

Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.

cot(-θ)/sec(-θ)

Verify each identity. cos² θ (1 + tan² θ) = 1

In Exercises 67–74, rewrite each expression in terms of the given function or functions. (sec x + csc x) (sin x + cos x) - 2 - cot x; tan x

Verify that each equation is an identity.

tan² α sin² α = tan² α + cos² α - 1

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

8. tan (-π/8)

In Exercises 47–54, use the figures to find the exact value of each trigonometric function. sin(θ/2)

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