Question

Find the value of $\frac{\tan {23}^{\circ }+\tan {22}^{\circ }}{1-\tan {23}^{\circ }.\tan {22}^{\circ }}$

Answer 1

GIven that:

$\frac{\tan {23}^{\circ }+\tan {22}^{\circ }}{1-\tan {23}^{\circ }.\tan {22}^{\circ }}$

We know that,

$\tan (\theta +\varphi )=\frac{\tan \theta +\tan \varphi }{1-\tan \theta \tan \varphi }$

So,

$\frac{\tan {23}^{\circ }+\tan {22}^{\circ }}{1-\tan {23}^{\circ }.\tan {22}^{\circ }}=\tan ({23}^{\circ }+{22}^{\circ })$

$\frac{\tan {23}^{\circ }+\tan {22}^{\circ }}{1-\tan {23}^{\circ }.\tan {22}^{\circ }}=\tan {45}^{\circ }$

$\because \tan \left({45}^{\circ }\right)=1$

$\therefore \frac{\tan {23}^{\circ }+\tan {22}^{\circ }}{1-\tan {23}^{\circ }.\tan {22}^{\circ }}=1$