Question

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

Answer 1

Now,

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

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

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

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