Prove the following trigonometric identities. (1+ ^2 )(1- )(1+ )=… - Vidyadip

Question

Prove the following trigonometric identities.
$(1+\tan^2\theta)(1-\sin\theta)(1+\sin\theta)=1$

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Answer

We have to prove $(1+\tan^2\theta)(1-\sin\theta)(1+\sin\theta)=1$ We know that, $\sin^2\theta+\cos^2\theta=1,$ $\sec^2\theta-\tan^2\theta=1$ So, $\text{L.H.S}=(1+\tan^2\theta)(1-\sin\theta)(1+\sin\theta)$ $=(1+\tan^2\theta)\{(1-\sin\theta)(1+\sin\theta)\}$ $=(1+\tan^2\theta)(1-\sin^2\theta)$ $=\sec^2\theta\cos^2\theta$ $=\frac{1}{\cos^2\theta}\cos^2\theta$$=1=\text{R.H.S}$
$\therefore\ \text{L.H.S}=\text{R.H.S}$

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