$\begin{align}  & \text{L}\text{.H}\text{.S} \\ & \frac{1-\cos \alpha }{\sin \alpha }=\frac{1-\cos \alpha }{\sin \alpha }\times \frac{1+\cos \alpha }{1+\cos \alpha } \\ & \frac{1-\cos \alpha }{\sin \alpha }=\frac{1-{{\cos }^{2}}\alpha }{\sin \alpha +\sin \alpha \cos \alpha } \\ & \frac{1-\cos \alpha }{\sin \alpha }=\frac{{{\sin }^{2}}\alpha }{\sin \alpha (1+\cos \alpha )} \\ & \frac{1-\cos \alpha }{\sin \alpha }=\frac{\sin \alpha }{1+\cos \alpha } \\ & \text{Divide both numerator and denominator by  }\sin \alpha  \\ & \frac{1-\cos \alpha }{\sin \alpha }=\frac{\frac{\sin \alpha }{\sin \alpha }}{\frac{1+\cos \alpha }{\sin \alpha }}=\frac{1}{\frac{1}{\sin \alpha }+\frac{\cos \alpha }{\sin \alpha }}=\frac{1}{\operatorname{cosec}\alpha +\cot \alpha } \\ & From\text{ }R.H.S \\ & \frac{1}{\text{cosec}\alpha \text{+cot}\alpha }=\frac{1}{\tfrac{1}{\sin \alpha }+\tfrac{\cos \alpha }{\sin \alpha }}=\frac{1}{\tfrac{1+\cos \alpha }{\sin \alpha }}=\frac{\sin \alpha }{1+\cos \alpha } \\ & \frac{1}{\text{cosec}\alpha \text{+cot}\alpha }=\frac{\sin \alpha \times (1-\cos \alpha )}{(1+\cos \alpha )(1-\cos \alpha )}=\frac{\sin \alpha (1-\cos \alpha )}{1-{{\cos }^{2}}\alpha } \\ & \frac{1}{\text{cosec}\alpha \text{+cot}\alpha }=\frac{\sin \alpha (1-\cos \alpha )}{{{\sin }^{2}}\alpha } \\ & \frac{1}{\text{cosec}\alpha \text{+cot}\alpha }=\frac{1-\cos \alpha }{\sin \alpha } \\\end{align}$