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

Question

Prove the following trigonometric identities.
$\frac{(1+\sin\theta)^2+(1-\sin\theta)^2}{2\cos^2\theta}=\frac{1+\sin^2\theta}{1-\sin^2\theta}$

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Answer

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

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