Question

Prove the trigonometric identity
$\sqrt{\frac{\csc A-1}{\csc A+1}}+\sqrt{\frac{\csc A+1}{\csc A-1}}=2\sec A$.

Answer 1

Formula:

${\csc }^{2}A-{\cot }^{2}A=1$

${\csc }^{2}A-1={\cot }^{2}A$

$LHS=\frac{\sqrt{\csc A-1}}{\sqrt{\csc A+1}}+\frac{\sqrt{\csc A+1}}{\sqrt{\csc A-1}}$

$=\frac{\sqrt{\csc A-1}\sqrt{\csc A-1}+\sqrt{\csc A+1}\sqrt{\csc A+1}}{\sqrt{\csc A+1}\sqrt{\csc A-1}}$

$=\frac{(\csc A-1)+(\csc A+1)}{\sqrt{{\csc }^{2}A-1}}$

$=\frac{2\csc A}{\cot A}$

$=\frac{2}{\sin A}\times \frac{\sin A}{\cos A}$

$=\frac{2}{\cos A}=2\sec A=RHS$

Hence proved.