Question

tan squared theta minus sin squared theta is equal to sin theta into sec square theta prove that tan squared theta minus sin square theta is equal to sin raise to power 4 theta into sec squared theta ​

Answer 1

Answer:

Prove that:

$tan^2\theta-sin^2\theta=sin^4\theta\:sec^2\theta$

$tan^2\theta-sin^2\theta$

$=\frac{sin^2\theta}{cos^2\theta}-sin^2\theta$

$=sin^2\theta(\frac{1}{cos^2\theta}-1)$

$=sin^2\theta(sec^2\theta-1)$

using

$\boxed{sec^2\theta-tan^2\theta=1}$

$=sin^2\theta\:tan^2\theta$

$=sin^2\theta\:(\frac{sin^2\theta}{cos^2\theta})$

$=sin^4\theta\:(\frac{1}{cos^2\theta})$

$=sin^4\theta\:sec^2\theta$

$\implies\boxed{tan^2\theta-sin^2\theta=sin^4\theta\:sec^2\theta}$