6. Trigonometric Identities and More Equations

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

2. csc 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

Use the even-odd identities to evaluate the expression.

$\(\cos\]\left\)(-\(\theta\[\right\))-\(\cos\]\theta\)$

Factor each trigonometric expression.

sec² θ - 1

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.

cot 18°

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.

(sin θ - cos θ) (csc θ + sec θ)

Perform each transformation. See Example 2.

Write cot x in terms of csc x.

For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.

cos² 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

Verify that each equation is an identity.

(sin² θ)/cos θ = sec θ - cos θ

Which equation results from applying the secant and tangent segment theorem to the figure?

Verify that each equation is an identity.

(cot² t - 1)/(1 + cot² t) = 1 - 2 sin² t

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

(1 - cos 2x)/sin 2x

Find sinθ.

csc θ = -8/5

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

cos 58° = ±√ (1 + cos 116°)/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.

1 + cot(-θ)/cot(-θ)

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