6. Trigonometric Identities and More Equations

Simplify the expression.

$\(\tan\)^2\(\theta\)-\(\sec\)^2\(\theta\)+1$

In Exercises 63–68, find the exact value of each expression. Do not use a calculator. csc 37° sec 53° - tan 53° cot 37°

Verify that each equation is an identity.

sin³ θ + cos³ θ = (cos θ + sin θ) (1 - cos θ sin θ)

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

(cos x sin 2x)/1 + cos 2x)

Perform each transformation. See Example 2.

Write cot x in terms of sin x.

Find values of the sine and cosine functions for each angle measure.

B, given cos 2B = 1/8 , 540° < 2B < 720°

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.

sin 162°

Verify that each equation is an identity.

1/(sec α - tan α) = sec α + tan α

Simplify each expression.

± √[(1 + cos (x/4))/2]

Verify that each equation is an identity.

sec² α - 1 = (sec 2α - 1)/(sec 2α + 1)

Verify that each equation is an identity.

2 cos³ x - cos x = (cos² x - sin² x)/sec x

Select the expression with the same value as the given expression.

$\(\sin\[\left\)(-38\(\degree\]\right\))$

Verify that each equation is an identity.

(csc θ + cot θ)/(tan θ + sin θ) = cot θ csc θ

Use the Pythagorean identities to rewrite the expression as a single term.

$\(\left\)(1+\(\csc\[\theta\]\right\))\(\left\)(1-\(\csc\[\theta\]\right\))$

Let csc x = -3. Find all possible values of (sin x + cos x)/sec x.