Question

Solve $\frac{\sin A}{\sec A+\tan A-1}+\frac{\cos A}{\text{cosec}A+\cot A-1}-1$

Answer 1

$\frac{\sin A}{\frac{1}{\cos A}+\frac{\sin A}{\cos A}-1}+\frac{\cos A}{\frac{1}{\sin A}+\frac{\cos A}{\sin A}-1}-1$

$\Rightarrow$ $\frac{\sin A\cos A}{\sin A-\cos A+1}+\frac{\sin A\cos A}{-\sin A+\cos A+1}-1$

$\Rightarrow \sin A\cos A\left[\frac{1+\cos A-\sin A+1-\cos A+\sin A}{(1+\cos A-\sin A)(1-\cos A+\sin A)}\right]-1$

$\Rightarrow \frac{\sin A\cos A\left(2\right)}{1-\cos A+\sin A+\cos A-{\cos }^{2}A+\cos A\sin A-\sin A+\sin A\cos A-{\sin }^{2}A}-1$

$\Rightarrow \frac{\sin A\cos A\left(2\right)}{1-\cos A+\sin A+\cos A-{\cos }^{2}A+\cos A\sin A-\sin A+\sin A\cos A-{\sin }^{2}A}-1$

$\Rightarrow \frac{2\sin A\cos A}{2\sin A\cos A}-1$

$\Rightarrow 1-1=0$

$\left[Answer\right]$