Question

Prove the following identities (1-17)

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

Answer 1

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