Question

If $\alpha +\beta =\frac{\pi }{4}$ the prove that $(1+tan\alpha )(1+tan\beta )=2$

Answer 1

Given: $\alpha +\beta =\frac{\pi }{4}$

$\tan (\alpha +\beta )=\tan \frac{\pi }{4}$

$\frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }=1$

$\Rightarrow \tan \alpha +\tan \beta =1-\tan \alpha \tan \beta$

$\Rightarrow \tan \alpha +\tan \beta +\tan \alpha \tan \beta =1$

$\Rightarrow \tan \alpha +\tan \beta +\tan \alpha \tan \beta =1$ by adding $1$ to both sides.

$\Rightarrow 1+tan\alpha +\tan \beta +\tan \alpha \tan \beta =1+1$

$\Rightarrow (1+tan\alpha )+\tan \beta (1+\tan \beta )=1+1$

$\therefore (1+tan\alpha )(1+tan\beta )=2$