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If $\tan \theta=\frac{1}{2}$ and $\tan \phi=\frac{1}{3}$, then the value of $\theta+\phi$ is
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1176 Upvotes
Verified Answer
The correct answer is:
$\frac{\pi}{4}$
$\frac{\pi}{4}$
Since, $\tan \theta=\frac{1}{2}$ and $\tan \phi=\frac{1}{3}$
$$
\begin{aligned}
&\because \tan (\theta+\phi)=\frac{\tan \theta+\tan \phi}{1-\tan \theta \times \tan \phi}=\frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{2} \times \frac{1}{3}} \\
&\Rightarrow \tan (\theta+\phi)=\frac{\frac{3+2}{6}}{\frac{6-1}{6}}=1=\tan \frac{\pi}{4}
\end{aligned}
$$
$$
\begin{aligned}
&\because \tan (\theta+\phi)=\frac{\tan \theta+\tan \phi}{1-\tan \theta \times \tan \phi}=\frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{2} \times \frac{1}{3}} \\
&\Rightarrow \tan (\theta+\phi)=\frac{\frac{3+2}{6}}{\frac{6-1}{6}}=1=\tan \frac{\pi}{4}
\end{aligned}
$$
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