Question
$$\tan \theta \cdot \tan \left( {\frac{\pi }{3} + \theta } \right) \cdot \tan \left( {\frac{\pi }{3} - \theta } \right)$$ is equal to
A.
$$\tan 2\theta $$
B.
$$\tan 3\theta $$
C.
$${\tan ^3}\theta $$
D.
None of these
Answer :
$$\tan 3\theta $$
Solution :
The expression $$ = \tan \theta \cdot \frac{{\sqrt 3 + \tan \theta }}{{1 - \sqrt 3 \tan \theta }} \cdot \frac{{\sqrt 3 - \tan \theta }}{{1 + \sqrt 3 \tan \theta }}$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{3\tan \theta - {{\tan }^3}\theta }}{{1 - 3{{\tan }^2}\theta }} = \tan 3\theta .$$