Question
Prove the following identities:
$\frac{1+\cos\theta-\sin^2\theta}{\sin\theta(1+\cos\theta)}=\cot\theta$
✓
Answer
$\text{LHS}=\frac{1+\cos\theta-\sin^2\theta}{\sin\theta(1+\cos\theta)}$
$=\frac{1+\cos\theta-\big(1-\cos^2\theta\big)}{\sin\theta(1+\cos\theta)}$
$=\frac{1+\cos\theta-1+\cos^2\theta}{\sin\theta(1+\cos\theta)}$
$=\frac{\cos\theta(1+\cos\theta)}{\sin\theta(1+\cos\theta)}$
$=\frac{\cos\theta}{\sin\theta}$
$=\cot\theta$
$=\text{R.H.S.}$
$\therefore\text{R.H.S.}=\text{L.H.S.}$
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