Question

Derive the identify $1+{\tan }^2\theta ={\sec }^2\theta$ from the identify ${\sin }^2\theta +{\cos }^2\theta =1$

Answer 1

Given that the identity ${\sin }^2\theta +{\cos }^2\theta =1.......\left(1\right)$

Prove that $1+{\tan }^2\theta ={\sec }^2\theta$

Proof:- By equation (1)

${\sin }^2\theta +{\cos }^2\theta =1$

Divide both side by ${\cos }^2\theta$ and we get,

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

$\Rightarrow {\tan }^2\theta +1={\sec }^2\theta$

$\Rightarrow 1+{\tan }^2\theta ={\sec }^2\theta$

Hence proved.