Prove the following identities (1-16)

Question:

Prove the following identities (1-16)

$\frac{2 \sin x \cos x-\cos x}{1-\sin x+\sin ^{2} x-\cos ^{2} x}=\cot x$

Solution:

$\mathrm{LHS}=\frac{2 \sin x \cos x-\cos x}{1-\sin x+\sin ^{2} x-\cos ^{2} x}$

$=\frac{\cos x(2 \sin x-1)}{2 \sin ^{2} x-\sin x} \quad\left(\because 1-\cos ^{2} x=\sin ^{2} x\right)$

$=\frac{\cos x(2 \sin x-1)}{\sin x(2 \sin x-1)}$

$=\cot x$

= RHS

Hence proved.

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