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Question
If \[\tan\theta = \frac{1}{2}\] and \[\tan\phi = \frac{1}{3}\], then the value of \[\tan\phi = \frac{1}{3}\] is
Options
- \[\frac{\pi}{6}\]
- \[\pi\]
0
- \[\frac{\pi}{4}\]
Solution
It is given that \[\tan\theta = \frac{1}{2}\] and \[\tan\phi = \frac{1}{3}\]
Now,
\[\tan\left( \theta + \phi \right) = \frac{\tan\theta + \tan\phi}{1 - \tan\theta\tan\phi}\]
\[ = \frac{\frac{1}{2} + \frac{1}{3}}{1 - \frac{1}{2} \times \frac{1}{3}}\]
\[ = \frac{\frac{5}{6}}{\frac{5}{6}}\]
\[ = 1\]
Hence, the correct answer is option D.
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