Question

State whether the following statement is true or false.

If $\alpha$ and $\beta$ are acute angles such that $\tan \alpha =\frac{m}{m+1}$ and $\tan \beta =\frac{1}{2m+1}$, then $\alpha +\beta =\frac{\pi }{4}$.

• A. True
• B. False

Answer 1

The correct option is A True

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

$\tan (\alpha +\beta )=\frac{\frac{m}{m+1}+\frac{1}{2m+1}}{1-\frac{m}{m+1}\times \frac{1}{2m+1}}$

$\tan (\alpha +\beta )=\frac{2m^2+2m+1}{2m^2+2m+1}=1=\tan \frac{\pi }{4}$

$⟹\alpha +\beta =\frac{\pi }{4}$