Question

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

Answer 1

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

multiplying and dividing by $cos\theta -sin\theta$

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

simplifying and using trigonometric identities

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

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