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. $3$
• E. $\frac{1}{2}$

Answer 1

Answer: (C)

Given, $\tan \theta =\frac{1}{2}$ and $\tan \varphi =\frac{1}{3}$
$\because \tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta }=\frac{2\times \frac{1}{2}}{1-\frac{1}{4}}$
$\tan 2\theta =\frac{1}{\frac{3}{4}}=\frac{4}{3}$
Now, $\tan (2\theta +\varphi )=\frac{\tan 2\theta +\tan \varphi }{1-\tan 2\theta \tan \varphi }=\frac{\frac{4}{3}+\frac{1}{3}}{1-\frac{4}{3}\times \frac{1}{3}}=\frac{\frac{5}{3}}{1-\frac{4}{9}}$
$=\frac{\frac{5}{3}}{\frac{5}{9}}=\frac{5}{3}\times \frac{9}{5}=3$