6. Trigonometric Identities and More Equations

Verify that each equation is an identity.

sin² x(1 + cot x) + cos² x(1 - tan x) + cot² x = csc² x

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.

csc x - cot x

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.

[1 - sin²(-θ)]/[1 + cot²(-θ)]

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

(1 + tan θ)² - 2 tan θ

For each expression in Column I, choose the expression from Column II that completes an identity.

6. sec² x = ____

II

A. sin ^2 x/cos ^2 x

B.1/(sec ^2 x)

C. sin (-x)

D. csc ^2 x-cot ^2 x + sin ^2 x

E. tan x

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

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. sin x + sin 2x

Find sinθ.

tan θ = -(√7)/2, sec θ > 0

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.

cot² θ(1 + tan² θ)

Verify that each equation is an identity.

(sec α - tan α)² = (1 - sin α)/(1 + sin α)

Verify that each equation is an identity.

sin² β (1 + cot² β) = 1

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

1/( sin α - 1) - 1/(sin α + 1)

Find the remaining five trigonometric functions of θ.

sin θ = -4/5, cos θ < 0

Verify each identity. csc θ - sin θ = cot θ cos θ

Identify the most helpful first step in verifying the identity.

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