Question

​​sin theta/1-cos theta = cosectheta +cot theta

Answer 1

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