Question

Prove the following identities (1-17)

${\sec }^4\mathrm{\theta }-{\sec }^2\mathrm{\theta }={\tan }^4\mathrm{\theta }+{\tan }^2\mathrm{\theta }$

Answer 1

$$\begin{gathered} \mathrm{LHS}={\sec }^4\theta -{\sec }^2\theta \\ ={\sec }^2\theta \left({\sec }^2\theta -1\right) \\ =\left({\tan }^2\theta +1\right)\left({\tan }^2\theta \right)\left(\because {\sec }^2\theta -{\tan }^2\theta =1\right) \\ ={\tan }^4\theta +{\tan }^2\theta \\ \mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}=\mathrm{RHS} \\ \mathrm{Hence}\mathrm{proved}. \end{gathered}$$