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Question
tan 20° + tan 40° + \[\sqrt{3}\] tan 20° tan 40° is equal to
Options
(a) \[\frac{\sqrt{3}}{4}\]
(b) \[\frac{\sqrt{3}}{2}\]
(c) \[\sqrt{3}\]
(d) 1
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Solution
(c) \[\sqrt{3}\]
\[\tan20° + \tan40° + \sqrt{3}\tan20°\tan40°\]
\[ = \tan 60°(1 - \tan20°\tan40°) + \tan60°\tan20°\tan40° \left[ \text{ Using } \tan60° = \frac{\tan20 + \tan40}{1 - \tan20\tan40} \text{ and } \tan60° = \sqrt{3} \right]\]
\[ = \tan60° - \tan60°\tan20°tan40° + \tan60°\tan20°\tan40°\]
\[ = \tan60° \]
\[ = \sqrt{3}\]
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