Prove that ^2 − ^2 = ^2 + ^2 - Vidyadip

Question

Prove that $\sec^2\theta − \cos^2\theta = \tan^2\theta + \sin^2\theta$

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Answer

$\text { L.H.S }=\sec ^2 \theta-\cos ^2 \theta$
$=1+\tan ^2 \theta-\cos ^2 \theta \quad \ldots \ldots . .\left[\because 1+\tan ^2 \theta=\sec ^2 \theta\right]$
$=\tan ^2 \theta+\left(1-\cos ^2 \theta\right)$
$=\tan ^2 \theta+\sin ^2 \theta \quad \ldots [\because \sin ^2 \theta+\cos ^2 \theta=1\therefore 1-\cos ^2 \theta=\sin ^2 \theta]$
$=\text { R.H.S }$
$\therefore \sec ^2 \theta-\cos ^2 \theta=\tan ^2 \theta+\sin ^2 \theta$

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