Prove the Identity : \[ \frac{2\sin x\cos x-\cos x} {1-\sin x+\sin^2 x-\cos^2 x} = \cot x \]

Solution:

\[ \frac{\cos x(2\sin x-1)} {1-\sin x+\sin^2 x-\cos^2 x} \]

Using \[ \cos^2 x=1-\sin^2 x \]

\[ = \frac{\cos x(2\sin x-1)} {1-\sin x+\sin^2 x-(1-\sin^2 x)} \]

\[ = \frac{\cos x(2\sin x-1)} {2\sin^2 x-\sin x} \]

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

\[ = \frac{\cos x}{\sin x} \]

\[ =\cot x \]

Hence proved.