Question

Show that $\tan (\frac{\pi }{4}+\theta )-\tan (\frac{\pi }{4}-\theta )=2\tan 2\theta$.

Answer 1

$\tan (\frac{\pi }{4}+\theta )-\tan (\frac{\pi }{4}-\theta )$

using trigonometric identity

$=\frac{1+\tan \theta }{1-\tan \theta }-\frac{1-\tan \theta }{1+\tan \theta }=\frac{{(1+\tan \theta )}^2-{(1-\tan \theta )}^2}{1-{\tan }^2\theta }$
$=\frac{4\tan \theta }{1-{\tan }^2\theta }=2\tan 2\theta$