Question

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

Answer 1

$LHS=$ ${\sec }^4\theta -{\sec }^2\theta$

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

$=(1+{\tan }^2\theta )(1+{\tan }^2\theta -1)$.........$\left(UsingTrigonometricIdentity\right)$

$=(1+{\tan }^2\theta ){\tan }^2\theta$

$={\tan }^2\theta +{\tan }^4\theta$

$=RHS$

Hence, proved.