Question

prove the identity :
Sec2θ(Sec2θ - 2) + 1 = tan4θ

Answer 1

$$\begin{gathered} \mathrm{To}\mathrm{Prove}:- \\ {\sec }^2\mathrm{\theta }\left({\sec }^2\mathrm{\theta }-2\right)+1={\tan }^4\mathrm{\theta } \\ \mathrm{LHS} \\ {\sec }^2\mathrm{\theta }\left({\sec }^2\mathrm{\theta }-2\right)+1 \\ ={\sec }^4\mathrm{\theta }-2{\sec }^2\mathrm{\theta }+1 \\ ={\left({\sec }^2\mathrm{\theta }-1\right)}^2\left(\mathrm{Since}{\left(\mathrm{a}-\mathrm{b}\right)}^2={\mathrm{a}}^2-2\mathrm{ab}+{\mathrm{b}}^2\right) \\ ={\left({\tan }^2\mathrm{\theta }\right)}^2\left(\mathrm{Since}1+{\tan }^2\mathrm{x}={\sec }^2\mathrm{x}\right) \\ ={\tan }^4\mathrm{\theta }=\mathrm{RHS} \\ \mathrm{Hence}\mathrm{Proved}. \end{gathered}$$