Question

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

• A. $\alpha +\beta =\pi /4$
• B. $\alpha -\beta =\pi /4$
• C. $\alpha +\beta =\pi /6$
• D. $\alpha -\beta =\pi /3$

Answer 1

The correct option is B $\alpha -\beta =\pi /4$

Given $\tan \alpha =\frac{m}{m-1}⟹m\tan \alpha -\tan \alpha =m⟹m=\frac{\tan \alpha }{\tan \alpha -1}\cdots \left(1\right)$

$\tan \beta =\frac{1}{2m-1}⟹2m\tan \beta -\tan \beta =1⟹m=\frac{1+\tan \beta }{2\tan \beta }\cdots \left(2\right)$

From $\left(1\right)$ and $\left(2\right)$

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

$⟹2\tan \alpha \tan \beta =-1+\tan \alpha -\tan \beta +\tan \alpha \tan \beta$

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

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

$⟹\tan (\alpha -\beta )=1$

$⟹\alpha -\beta =\frac{\pi }{4}$