6. Trigonometric Identities and More Equations

Verify that each equation is an identity.

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

Verify that each equation is an identity.

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

Verify that each equation is an identity.

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

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

Verify each identity. cos θ sec θ/cot θ= tan θ

Verify each identity. cos² θ (1 + tan² θ) = 1

In Exercises 1–60, verify each identity. cot² t /csc t = csc t - sin t

If θ is an acute angle and sin θ = (2√7) / 7, use the identity sin²θ + cos²θ = 1 to find cos θ.

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

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

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

In Exercises 67–74, rewrite each expression in terms of the given function or functions. (sec x + csc x) (sin x + cos x) - 2 - cot x; tan x

In Exercises 35–38, use the power-reducing formulas to rewrite each expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1. 6 sin⁴ x

In Exercises 35–38, use the power-reducing formulas to rewrite each expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1. sin² x cos² x

In Exercises 39–46, use a half-angle formula to find the exact value of each expression. tan(7𝝅/8)