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  4. If tan θ = (1/2) and tan φ = ( 1/ 3) , then tan (2θ + φ)is equal to

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

KEAMKEAM 2016Trigonometric Functions

Solution:

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