Question

Prove that
$2{\sec }^2\theta -{\sec }^4\theta -2{\csc }^2\theta +{\csc }^4\theta ={\cot }^4\theta -{\tan }^4\theta$

Answer 1

$\begin{array}{c}2{\sec }^2\theta -{\sec }^4\theta -2{\csc }^2\theta +{\csc }^4\theta ={\cot }^4\theta -{\tan }^4\theta \\ 2\left(1+{\tan }^2\theta \right)-{\left(1+{\tan }^2\theta \right)}^2-2\left(1+{\cot }^2\theta \right)+{\left(1+{\cot }^2\theta \right)}^2\\ =2+2{\tan }^2\theta -\left(1+{\tan }^4\theta +2{\tan }^2\theta \right)-\left(2+2{\cot }^2\theta \right)+1+{\cot }^4\theta +2{\cot }^2\theta \\ =2+2{\tan }^2\theta -1-{\tan }^4\theta -2{\tan }^2\theta -2-2{\cot }^2\theta +1+{\cot }^4\theta +2{\cot }^2\theta \\ ={\cot }^4\theta -{\tan }^4\theta \\ \end{array}$

R.H.S

Hence, proved.