Question

Prove the identify : ${\sec }^2\theta \left({\sec }^2\theta -2\right)+1={\tan }^4\theta .$

Answer 1

L.H.S
${\sec }^2\theta ({\sec }^2\theta -2)+1$

$\Rightarrow {\sec }^4\theta -2{\sec }^2\theta +1$

$\Rightarrow ({\sec }^2\theta -1{)}^2$

We know that

${\sec }^{2}x+{\tan }^{2}x=1$

Therefore,

$\Rightarrow ({\tan }^2\theta {)}^2$

$\Rightarrow {\tan }^4\theta$

R.H.S

Hence, proved.