Question

If tan $\theta =\frac{1}{2}$ and $tan\varphi =\frac{1}{3}$, then the value of $\theta +\varphi$ is

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

Answer 1

Answer: (C)

$\frac{\pi }{4}$

$\tan \theta =\frac{1}{2},\tan \varphi =\frac{1}{3}$

$\tan (\theta +\varphi )=\frac{\tan \theta +\tan \varphi }{1-\tan \theta \tan \varphi }$

$=\frac{\frac{1}{2}+\frac{1}{3}}{1-\left(\frac{1}{2}\right)\left(\frac{1}{3}\right)}$

$=\frac{\frac{5}{6}}{\frac{5}{6}}$

$\therefore \tan (\theta +\varphi )=1$

$\Rightarrow \theta +\varphi =\frac{\pi }{4}$