Question

Prove the following identities (1-16)

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

Answer 1

$$\begin{gathered} \mathrm{LHS}=\frac{2\sin x\cos x-\cos x}{1-\sin x+{\sin }^{2}x-{\cos }^{2}x} \\ =\frac{\cos x(2\sin x-1)}{2{\sin }^{2}x-\sin x}\left(\because 1-{\cos }^{2}x={\sin }^{2}x\right) \\ =\frac{\cos x(2\sin x-1)}{\sin x\left(2\sin x-1\right)} \\ =\cot x \\ =\mathrm{RHS} \\ \mathrm{Hence}\mathrm{proved}. \end{gathered}$$