Question

Simplify `("csc"^2x)/(1 + tan^2x)` ?

Answer 1

See below

Explanation:

The easiest trigonometric ratio that it could simplify to is `cot^2x`

Step1: Convert everything to `sinx` and `cosx`

`csc^2x` can be converted into `1/sin^2x` according to the reciprocal identities and `1+tan^2x` can be converted into `sec^2x` according to the Pythagorean identities, and then convert that into `1/cos^2x` because its the reciprocal.

Step2: Plug these converted identities back into the equation

`(1/sin^2x)/(1/cos^2x)`

when dividing by a fraction, you flip the bottom fraction and multiply.

Answer

`(1/sin^2x) ⋅ (cos^2x/1) = cos^2x/sin^2x = cot^2x`

according to the Quotient identities.