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Question
If tan 69° + tan 66° − tan 69° tan 66° = 2k, then k =
Options
−1
- \[\frac{1}{2}\]
- \[- \frac{1}{2}\]
None of these
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Solution
\[\tan135^\circ = \tan(90^\circ + 45^\circ)\]
\[ = - \tan45^\circ\]
\[ = - 1\]
\[Or, \tan(69^\circ + 66^\circ) = \frac{\tan69^\circ + \tan66^\circ}{1 - \tan69^\circ \tan66^\circ}\]
\[ \Rightarrow - 1 = \frac{\tan69^\circ + \tan66^\circ}{1 - \tan69^\circ \tan66^\circ}\]
\[ \Rightarrow \tan69^\circ + \tan66^\circ - \tan69^\circ\tan66^\circ = - 1\]
Therefore,
\[2k = - 1\]
\[ \Rightarrow k = \frac{- 1}{2}\]
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