6. Trigonometric Identities and More Equations

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.

1 - 1/sec² x

Concept Check Suppose that sec θ = (x+4)/x.

Find an expression in x for tan θ.

Verify that each equation is an identity.

tan α/sec α = sin α

Perform each transformation. See Example 2.

Write cot x in terms of sin x.

Verify that each equation is an identity.

(tan² α + 1)/ sec α = sec α

Perform each transformation. See Example 2.

Write cot x in terms of csc x.

Verify that each equation is an identity.

sin² β (1 + cot² β) = 1

Verify that each equation is an identity.

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

Verify that each equation is an identity.

sin² α + tan² α + cos² α = sec² α

Verify that each equation is an identity.

(sin 2x)/(sin x) = 2/sec 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.

tan θ cos θ

Verify that each equation is an identity.

(sin² θ)/cos θ = sec θ - cos θ

Verify that each equation is an identity.

(2 tan B)/(sin 2B) = sec² B

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 cos θ tan θ

Verify that each equation is an identity.

sec⁴ x - sec² x = tan⁴ x + tan² x