Question

Prove the following identities:

${\displaystyle \frac{1}{\sin A+\cos A}+\frac{1}{\sin A-\cos A}=\frac{2\sin A}{1-2{\cos }^{2}A}}$

Answer 1

L.H.S. = ${\displaystyle \frac{1}{\sin A+\cos A}+\frac{1}{\sin A-\cos A}}$

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

= ${\displaystyle \frac{2\sin A}{{\sin }^{2}A-{\cos }^{2}A}}$

= ${\displaystyle \frac{2\sin A}{1-{\cos }^{2}A-{\cos }^{2}A}}$     ...(∵ sin²A = 1 – cos²A)

= ${\displaystyle \frac{2\sin A}{1-2{\cos }^{2}A}}$

= R.H.S.