Question

If $\tan \theta =\frac{1}{2}$ and $\tan \varphi =\frac{1}{3}$, then $\tan (2\theta +\varphi )$ is equal to

• A. $\frac{3}{4}$
• B. $\frac{4}{3}$
• C. $\frac{1}{3}$
• D. $\frac{1}{2}$

Answer 1

The correct option is D $\frac{1}{2}$

Given, $\tan \theta =\frac{1}{2};\tan \varphi =\frac{1}{3}$
$\because \tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta }$
$=\frac{2\times \frac{1}{2}}{1-\frac{1}{4}}=\frac{4}{3}$
Now, $\tan (2\theta +\varphi )=\frac{\tan 2\theta +\tan \varphi }{1-\tan 2\theta \tan \varphi }$
$=\frac{\frac{4}{3}+\frac{1}{3}}{1-\frac{4}{3}\times \frac{1}{3}}=3$