Prove the following identities. ^4 (1- ^4 )-2 ^2 =1 - Vidyadip

Question

Prove the following identities.
$\sec ^4 \theta\left(1-\sin ^4 \theta\right)-2 \tan ^2 \theta=1$

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Answer

L.H.S =$\sec ^4 \theta\left(1-\sin ^4 \theta\right)-2 \tan ^2 \theta$
= $1/\cos^4 \theta [1 - (\sin^2 \theta)^2]- 2 xx (\sin^2 \theta)/(\cos^2 \theta)$
= $1/(\cos^4 \theta) (1 + \sin^2 \theta) (1 - \sin^2 \theta) - 2 (\sin^2 \theta)/(\cos^2 \theta)$
= $1/(\cos^4 \theta) xx \cos^2 \theta (1 + \sin^2 \theta) - 2 (\sin^2 \theta)/(\cos^2 \theta)$
= $(1 + \sin^2 \theta)/(\cos^2 \theta) - (2\sin^2 \theta)/(\cos^2 \theta)$
= $(1 + \sin^2 \theta - 2\sin^2 \theta)/(\cos^2 \theta)$
= $(1 - \sin^2 \theta)/(\cos^2 \theta)$
= $(\cos^2 \theta)/(\cos^2 \theta)$
L.H.S = R.H.S
$\therefore \sec ^4 \theta\left(1-\mid \sin ^4 \theta\right)-2 \tan ^2 \theta=1$

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