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Question
Prove that:
\[\tan\frac{\pi}{12} + \tan\frac{\pi}{6} + \tan\frac{\pi}{12}\tan\frac{\pi}{6} = 1\]
Solution
\[ \frac{\pi}{12} = 15^\circ, \frac{\pi}{6} = 30^\circ\]
\[\text{ We know that }45^\circ = 15^\circ + 30^\circ\]
Therefore,
\[\tan\left( 45^\circ \right) = \tan\left( 15^\circ + 30^\circ \right)\]
\[ \Rightarrow 1 = \frac{\tan15^\circ + \tan30^\circ}{1 - \tan15^\circ \tan30^\circ}\]
\[ \Rightarrow 1 - \tan15^\circ \tan30^\circ = \tan15^\circ + \tan30^\circ \]
\[ \Rightarrow 1 = \tan15^\circ + \tan30^\circ + \tan15^\circ \tan30^\circ\]
\[ \Rightarrow \tan15^\circ + \tan30^\circ + \tan15^\circ\tan30^\circ = 1\]
Hence proved .
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