Question

If $\tan \alpha =\frac{m}{m+1}$ and $\tan \beta =\frac{1}{2m+1}$, then $\alpha +\beta =$.

• A. $\frac{\pi }{4}$
• B. $\frac{\pi }{2}$
• C. $\frac{3\pi }{4}$
• D. $\frac{3\pi }{2}$

Answer 1

The correct option is A $\frac{\pi }{4}$

$\tan \alpha =\frac{m}{m+1}$ $\tan \beta =\frac{1}{2m+1}$

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

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

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

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

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

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

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

$=\frac{\pi }{4}$.