Question

Prove the following identity:
x. $\frac{1}{\cos A+\sin A–1}+\frac{1}{\cos A+\sin A+1}=cosecA+\sec A$

Answer 1

Proving LHS = RHS.

First we consider Left Hand Side (LHS),

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

By taking LCM we get,

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

We know that, ${(a+b)}^2=a^2+2ab+b^2$

$=\frac{2(\cos A+sinA)}{{\cos }^{2}A+{\sin }^{2}A+2\cos A\sin A–1}$

$=\frac{2(\cos A+\sin A)}{2\cos A\sin A}$

$=\frac{\cos A+\sin A}{\cos A\sin A}$

$=\frac{\cos A}{\cos A\sin A}+\frac{\sin A}{\cos A\sin A}$

$=cosecA+secA$.

Then, Right Hand Side$=cosecA+\sec A$.

Therefore, LHS = RHS.

Hence proved.