Question

Show that $\frac{\sin \theta +\cos \theta }{\sin \theta -\cos \theta }+\frac{\sin \theta -\cos \theta }{\sin \theta +\cos \theta }$ $=-2\sec 2\theta$.

Answer 1

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

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

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

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

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

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

$=-2\sec 2\theta$