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Question
Prove that:
tan 36° + tan 9° + tan 36° tan 9° = 1
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Solution
\[\text{ We know that }36^\circ + 9^\circ = 45^\circ\]
Therefore,
\[ \tan\left( 36^\circ + 9^\circ \right) = \tan45^\circ\]
\[ \Rightarrow \frac{\tan36^\circ + \tan9^\circ}{1 - \tan36^\circ \tan9^\circ} = 1\]
\[ \Rightarrow \tan36^\circ + \tan9^\circ = 1 - \tan36^\circ \tan9^\circ\]
\[ \Rightarrow \tan36^\circ + \tan9^\circ + \tan36^\circ \tan9^\circ = 1\]
Hence proved.
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