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If $\tan \theta=\frac{1}{2}$ and $\tan \varphi=\frac{1}{3}$, then what is the value of $?
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Verified Answer
The correct answer is:
$\frac{\pi}{4}$
\tan \theta=\frac{1}{2} \\ & \tan \phi=\frac{1}{3} \end{aligned}$
$\tan (\theta+\phi)=\frac{\tan \theta+\tan \phi}{1-\tan \theta \tan \phi}$
$=\frac{\frac{1}{2}+\frac{1}{3}}{1-\left(\frac{1}{2}\right)\left(\frac{1}{3}\right)}=\frac{\frac{5}{6}}{\frac{5}{6}}=1$
$\therefore \theta+\phi=\tan ^{-1}(1)=\frac{\pi}{4}$
$\tan (\theta+\phi)=\frac{\tan \theta+\tan \phi}{1-\tan \theta \tan \phi}$
$=\frac{\frac{1}{2}+\frac{1}{3}}{1-\left(\frac{1}{2}\right)\left(\frac{1}{3}\right)}=\frac{\frac{5}{6}}{\frac{5}{6}}=1$
$\therefore \theta+\phi=\tan ^{-1}(1)=\frac{\pi}{4}$
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