Establish the identity. 1 − sec θ / tan θ − tan θ / 1 − sec θ = 2 cot θ

Question

Write the left side of the expression with a common denominator. Do not expand the numerator.

? / tan θ(1 - sec θ).

Expand and simplify the numerator by rewriting without any parentheses.

? - tan²θ / tan θ(1 - sec θ).

Apply an appropriate Pythagorean Identity to simplify the numerator of the expression from the previous step.

? / tan θ(1 - sec θ). (Do not factor.)

Factor the numerator and cancel out the like terms of the expression from the previous step. Do not apply any trigonometric identity.

? / ? (Simplify your answers.)

The fraction from the previous step then simplifies to 2 cot θ using​ what?

A. Reciprocal Identity.

B. Even-Odd Identity.

C. Cancellation Identity.

D. Quotient Identity.

E. Pythagorean Identity.

Answer 1

This is all very straight forward. Just follow each step like this:

In the last line, the reciprocal of 1/tan(θ) is cot(θ)

Answer 2

(1 − secθ)/tan θ − tanθ/(1 − secθ) = [(1 - secθ)² -tan²θ]/[tanθ(1 - secθ)] = (1 - 2secθ + sec²θ - tan²θ)/[(1 - secθ)tanθ] = [2(1 - secθ)]/[(1 - secθ)tanθ] = 2/tanθ = 2cotθ

Answer 3

desmos.com/calculator/53rvffiibu

The above graph shows that indeed the left side of the identity is identical to the right side.