Prove the following identities: (1- ) + ^2 ( - ) =( + ) - Vidyadip

Question

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

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Answer

$\text{L.H.S.}=\frac{\cos\theta}{(1-\tan\theta)}+\frac{\sin^2\theta}{(\cos\theta-\sin\theta)}$
$=\frac{\cos\theta}{\Big(1-\frac{\sin\theta}{\cos\theta}\Big)}+\frac{\sin^2\theta}{(\cos\theta-\sin\theta)}$
$=\frac{\cos^2\theta}{(\cos\theta-\sin\theta)}+\frac{\sin^2\theta}{(\cos\theta-\sin\theta)}$
$=\frac{\cos^2\theta-\sin^2\theta}{\big(\cos\theta-\sin\theta\big)}$
$=\frac{(\cos\theta-\sin\theta)(\cos\theta+\sin\theta)}{(\cos\theta-\sin\theta)}$
$=(\cos\theta+\sin\theta)$
$=\text{R.H.S.}$
$\therefore\text{R.H.S.}=\text{L.H.S.}$

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