$\begin{align}  & \text{L}\text{.H}\text{.S} \\ & \frac{1}{\sin \alpha +\cos \alpha }=\frac{{{\sin }^{2}}\alpha +{{\cos }^{2}}\alpha }{\sin \alpha +\cos \alpha }\left| note:{{\sin }^{2}}\alpha +{{\cos }^{2}}\alpha =1 \right. \\ & \text{Divide the numerator and denominator by }\sin \alpha \cos \alpha  \\ & \frac{1}{\sin \alpha +\cos \alpha }=\frac{\frac{{{\sin }^{2}}\alpha +{{\cos }^{2}}\alpha }{\sin \alpha \cos \alpha }}{\frac{\sin \alpha +\cos \alpha }{\sin \alpha \cos \alpha }}=\frac{\frac{{{\sin }^{2}}\alpha }{\sin \alpha \cos \alpha }+\frac{{{\cos }^{2}}\alpha }{\sin \alpha \cos \alpha }}{\frac{\sin \alpha }{\sin \alpha \cos \alpha }+\frac{\cos \alpha }{\sin \alpha \cos \alpha }} \\ & \frac{1}{\sin \alpha +\cos \alpha }=\frac{\frac{\sin \alpha }{\cos \alpha }+\frac{\cos \alpha }{\sin \alpha }}{\frac{1}{\cos \alpha }+\frac{1}{\sin \alpha }}=\frac{\tan \alpha +\cot \alpha }{\sec \alpha +\operatorname{cosec}\alpha } \\ & \frac{1}{\sin \alpha +\cos \alpha }=\frac{\tan \alpha +\cot \alpha }{\sec \alpha +\operatorname{cosec}\alpha } \\ &  \\ & \text{From R}\text{.H}\text{.S} \\ & \frac{\tan \alpha +\cot \alpha }{\sec \alpha +\operatorname{cosec}\alpha }=\frac{\frac{\sin \alpha }{\cos \alpha }+\frac{\cos \alpha }{\sin \alpha }}{\frac{1}{\cos \alpha }+\frac{1}{\sin \theta }}=\frac{\frac{{{\sin }^{2}}\alpha +{{\cos }^{2}}\alpha }{\sin \alpha \cos \alpha }}{\frac{\sin \alpha +\cos \alpha }{\sin \alpha \cos \alpha }} \\ & \frac{\tan \alpha +\cot \alpha }{\sec \alpha +\operatorname{cosec}\alpha }=\frac{{{\sin }^{2}}\alpha +{{\cos }^{2}}\alpha }{\sin \alpha \cos \alpha }\times \frac{\sin \alpha \cos \alpha }{\sin \alpha +\cos \alpha } \\ & \frac{\tan \alpha +\cot \alpha }{\sec \alpha +\operatorname{cosec}\alpha }=\frac{{{\sin }^{2}}\alpha +{{\cos }^{2}}\alpha }{\sin \alpha +\cos \alpha } \\ & \frac{\tan \alpha +\cot \alpha }{\sec \alpha +\operatorname{cosec}\alpha }=\frac{1}{\sin \alpha +\cos \alpha } \\\end{align}$