Question

$\frac{\cot A-\cos A}{\cot A+\cos A}=\frac{\mathrm{cosec}A-1}{\mathrm{cosec}A+1}$

Answer 1

$$\begin{gathered} \frac{\cot A-\cos A}{\cot A+\cos A} \\ =\frac{{\displaystyle \frac{\cos A}{\sin A}}-\cos A}{{\displaystyle \frac{\cos A}{\sin A}}+\cos A} \\ =\frac{\cos A\left({\displaystyle \frac{1}{\sin A}}-1\right)}{\cos A\left({\displaystyle \frac{1}{\sin A}}+1\right)} \\ =\frac{\mathrm{cosec}A-1}{\mathrm{cosec}A+1}\left(\mathrm{cosec}A=\frac{1}{\sin A}\right) \end{gathered}$$