Question

Is ${\tan }^{2}x=se{c}^{2}x-1$ an identity?

Answer 1

Prove the identity :

We know that, ${\sin }^{2}x+{\cos }^{2}x=1$

Divide the equation by ${\cos }^{2}x$.

$\Rightarrow$ $\frac{{\sin }^{2}x}{{\cos }^{2}x}+\frac{{\cos }^{2}x}{{\cos }^{2}x}=\frac{1}{{\cos }^{2}x}$

Since, $\frac{\sin x}{\cos x}=\tan x,$and $\frac{1}{\cos x}=secx$

$\Rightarrow$${\tan }^{2}x+1=se{c}^{2}x$

rearranging the terms,

$\Rightarrow$${\sec }^2\mathrm{x}-1={\tan }^2\mathrm{x}$

Hence, ${\tan }^{2}x={\sec }^{2}x-1$ is an identity.