6. Trigonometric Identities and More Equations

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.

csc θ - sin θ

In Exercises 67–74, rewrite each expression in terms of the given function or functions. $\frac{1}{1-\cos x}-\frac{\cos x}{1+\cos x}$; $\csc x$

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

cos β(sec β + csc β)

Advanced methods of trigonometry can be used to find the following exact value.

sin 18° = (√5 - 1)/4

(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.

cos 18°

Verify that each equation is an identity.

(1 - cos θ)/(1 + cos θ) = 2 csc² θ - 2 csc θ cot θ - 1

Simplify each expression.

sin 158.2°/(1 + cos 158.2°)

Use the given information to find each of the following.

cos θ, given cos 2θ = 1/2 and θ terminates in quadrant II

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 300°

Identify the basic trigonometric function graphed, and determine whether it is even or odd.

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Find the remaining five trigonometric functions of θ.

cos θ = -1/4, sin θ > 0

Factor each trigonometric expression.

(tan x + cot x)² - (tan x - cot x)²

Advanced methods of trigonometry can be used to find the following exact value.

sin 18° = (√5 - 1)/4

(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.

csc 18°

Use a calculator to determine whether each statement is true or false. A true statement may lead to results that differ in the last decimal place due to rounding error. cos 40° = 2 cos 20°

Determine whether the positive or negative square root should be selected.

sin (-10°) = ± √[(1 - cos (-20°))/2]