Simplify the following expression: A1 - A + A1 - A - A

Answer: Option D

$$\begin{array}{rl}& \frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\cot A}-\sin A\\ & =\frac{\cos A}{1-\frac{\sin A}{\cos A}}+\frac{\sin A}{1-\frac{\cos A}{\sin A}}-\sin A\\ & =\frac{{\cos }^{2}A}{\cos A-\sin A}+\frac{{\sin }^{2}A}{\sin A-\cos A}-\sin A\\ & =\frac{{\cos }^{2}A}{\cos A-\sin A}-\frac{{\sin }^{2}A}{\cos A-\sin A}-\sin A\\ & =\frac{{\cos }^{2}A-{\sin }^{2}A}{\cos A-\sin A}-\sin A\\ & =\frac{{\cos }^{2}A-{\sin }^{2}A-\sin A\cos A+{\sin }^{2}A}{\cos A-\sin A}\\ & =\frac{{\cos }^{2}A-\sin A\cos A}{\cos A-\sin A}\\ & =\frac{\cos A\left(\cos A-\sin A\right)}{\cos A-\sin A}\\ & =\boxed{\cos A}\\ & \\ & \mathbf{Alternative}:-\\ & \text{You can put any }A\text{ value and satisfied equation,}\\ & \text{let }A=135^{\circ }\\ & \frac{\cos {135}^{\circ }}{1-\tan {135}^{\circ }}+\frac{\sin {135}^{\circ }}{1-\cot {135}^{\circ }}-\sin 135^{\circ }\\ & =\frac{-\frac{1}{\sqrt{2}}}{2}+\frac{-\frac{1}{\sqrt{2}}}{2}-\frac{1}{\sqrt{2}}\\ & =-\frac{1}{\sqrt{2}}\\ & \text{Put in option (D) }\cos 135^{\circ }=-\frac{1}{\sqrt{2}}\text{ satisfied}\\ & \therefore \text{Option D is correct}\text{.}\end{array}$$