Question

cosA/(1-tanA)+sinA/(1-cotA)=sinA+cosA

Answer 1

$$\begin{array}{l} \frac{\cos A}{(1 - \tan A)} + \frac{\sin A}{(1 - \cot A)} = \\ = \frac{\cos^{2} A}{(\cos A - \sin A)} + \frac{\sin^{2} A}{(\sin A - \cos A)} = \\ = \frac{\cos^{2} A}{(\cos A - \sin A)} - \frac{\sin^{2} A}{(\cos A - \sin A)} = \\ = \frac{\cos^{2} A - \sin^{2} A}{(\cos A - \sin A)} = \frac{(\cos A - \sin A)(\cos A + \sin A)}{(\cos A - \sin A)} = \\ = \cos A + \sin A = \sin A + \cos A \Rightarrow \\ \Rightarrow \sin A + \cos A = \sin A + \cos A \Rightarrow \text{Proved}. \end{array}$$