Question

If $\alpha$ and $\beta$ are acute angle such that $\tan \alpha =\frac{m}{m+1}$ and $\tan \beta =\frac{1}{2m+1}$. Then find m if
$\alpha +\beta =\frac{\pi }{m}$

Answer 1

$\tan \left(\alpha +\beta \right)=\frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }$

$=\frac{\frac{m}{m+1}+\frac{1}{2m+1}}{1-\frac{m}{m+1}\times \frac{1}{2m+1}}$

$=\frac{m\left(2m+1\right)+\left(m+1\right)}{\left(m+1\right)\left(2m+1\right)-m}$

$=\frac{2m^2+2m+1}{2m^2+2m+1}$

$=1$

$\Rightarrow \alpha +\beta =\frac{\pi }{4}$

On comparing it with $\alpha +\beta =\frac{\pi }{m}$

we get $m=4$