Question

Given that $\tan \, \alpha = m/(m + 1), \tan \, \beta = 1/(2m+ 1),$ then what is the value of $\alpha + \beta$ ?

• A. 0
• B. $\frac{\pi}{4}$
• C. $\frac{\pi}{6}$
• D. $\frac{\pi}{3}$

Answer 1

Answer: (B)

As given, $\tan \alpha = \frac{m}{m + 1}$
and $\tan \, \beta = \frac{1}{ 2m + 1}$
$\tan (\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 + \tan \alpha \tan \beta}$
$= \frac{\frac{m}{m+1} + \frac{1}{2m+1}}{1- \frac{m}{m+1} \times\frac{1}{2m+1}} = \frac{m\left(2m+1\right)+ \left(m+1\right)}{\left(m+1\right)\left(2m+1\right) -m} $
$= \frac{2m^{2} + 2m +1}{2m^{2} + 2m+1} = 1$
So, $ \alpha+\beta = \frac{\pi}{4}$