Question

Solve
$(\csc A-\sin A)(\sec A-\cos A)(\tan A+\cot A)$

Answer 1

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

$=(\frac{1-{\sin }^{2}A}{\sin A})(\frac{1-{\cos }^{2}A}{\cos A})(\frac{1+{\tan }^{2}A}{\tan A})$

$=\frac{{\cos }^{2}A}{\sin A}\times \frac{{\sin }^{2}A}{\cos A}\times \frac{{\text{sec}}^{2}A}{\tan A}=\frac{1}{\sin A\cos A}\times \frac{\cos A}{\sin A{\cos }^{2}A}={\text{cosec}}^{2}A{\text{sec}}^{2}A$