Question

Prove the following identity:
$\frac{1}{\sin \theta +\cos \theta }+\frac{1}{\sin \theta -\cos \theta }=\frac{2\sin \theta }{1-2{\cos }^2\theta }$

Answer 1

$\frac{1}{\sin \theta +\cos \theta }+\frac{1}{\sin \theta -\cos \theta }$

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

$=\frac{2\sin \theta }{{\sin }^2\theta -{\cos }^2\theta }$

$=\frac{2\sin \theta }{(1-{\cos }^2\theta )-{\cos }^2\theta }$

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