Question

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

(a) $\frac{\mathrm{\pi }}{6}$ (b) $\mathrm{\pi }$ (c) 0 (d) $\frac{\mathrm{\pi }}{4}$

Answer 1

It is given that $\tan \theta =\frac{1}{2}$ and $\tan \varphi =\frac{1}{3}$.

Now,

$$\begin{gathered} \tan \left(\theta +\varphi \right)=\frac{\tan \theta +\tan \varphi }{1-\tan \theta \tan \varphi } \\ =\frac{{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{3}}}{1-{\displaystyle \frac{1}{2}}\times {\displaystyle \frac{1}{3}}} \\ =\frac{{\displaystyle \frac{5}{6}}}{{\displaystyle \frac{5}{6}}} \\ =1 \end{gathered}$$

$\therefore \theta +\varphi =\frac{\mathrm{\pi }}{4}\left(\tan \frac{\mathrm{\pi }}{4}=1\right)$

Hence, the correct answer is option D.