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Question
Prove that \[\frac{\tan 69^\circ + \tan 66^\circ}{1 - \tan 69^\circ \tan 66^\circ} = - 1\].
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Solution
\[\text{ LHS }= \frac{\tan69^\circ + \tan66^\circ}{1 - \tan69^\circ\tan66^\circ}\]
\[ = \tan\left( 69^\circ + 66^\circ \right) \left[\text{ Using the formula }\frac{\tan A + \tan B}{1 - \tan A\tan B} = \tan\left( A + B \right) \right]\]
\[ = \tan135^\circ\]
\[ = \tan\left( 180^\circ - 45^\circ \right)\]
\[ = - \tan45^\circ \left[ \tan\left( 180 - A \right) = - \tan A \right]\]
\[ = - 1\]
= RHS
Hence proved.
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