Question

stion Prove the following identities (1-16)
${\sec }^{4}x-{\sec }^{2}x={\tan }^{4}x+{\tan }^{2}x$

Answer 1

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