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

Jan 25, 2018

See below

Explanation:

The easiest trigonometric ratio that it could simplify to is ${\cot}^{2} x$

Step1: Convert everything to $\sin x$ and $\cos x$

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

Step2: Plug these converted identities back into the equation

$\frac{\frac{1}{\sin} ^ 2 x}{\frac{1}{\cos} ^ 2 x}$

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

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