Question

If $\tan \theta + \tan 4\theta + \tan 7\theta = \tan \theta \tan 4\theta \tan 7\theta $ then the general solution is :-
$
  {\text{A}}{\text{.}}\theta = \dfrac{{n\pi }}{4} \\
  {\text{B}}{\text{.}}\theta = \dfrac{{n\pi }}{{12}} \\
  {\text{C}}{\text{.}}\theta = \dfrac{{n\pi }}{6} \\
  {\text{D}}{\text{. None of these}} \\
$

Answer 1

Hint : Use the formula $\tan (a + b + c)$ and consider $a,b,c$ as $\theta ,4\theta ,7\theta $. Here remembering the trigonometric formula is a key point.

The given equation is
$\tan \theta + \tan 4\theta + \tan 7\theta = \tan \theta \tan 4\theta \tan 7\theta $
After transposing we get,
$\tan \theta + \tan 4\theta + \tan 7\theta - \tan \theta \tan 4\theta \tan 7\theta = 0{\text{ }}............{\text{(i)}}$
As we know
$\tan (a + b + c) = \dfrac{{\tan a + \tan b + \tan c - \tan a\tan b\tan c}}{{1 - \tan a\tan b - \tan a\tan c - \tan b\tan c}}{\text{ }}...........{\text{(ii)}}$
Use the above equation for the given equation we get,
$\tan (\theta + 4\theta + 7\theta ) = \dfrac{{\tan \theta + \tan 4\theta + \tan 7\theta - \tan \theta \tan 4\theta \tan 7\theta }}{{1 - \tan \theta \tan 4\theta - \tan \theta \tan 7\theta - \tan 7\theta \tan 4\theta }}{\text{ = }}\tan (12\theta ){\text{ }}...........{\text{(iii)}}$
But from equation (i) we say the numerator of equation (iii) is zero.
Therefore,
$
  {\text{tan(12}}\theta ) = 0 \\
  {\text{12}}\theta = n\pi \\
  \theta {\text{ = }}\dfrac{{n\pi }}{{12}} \\
$
Hence the correct option is B.

Note :- In these types of questions of finding general values of angles we have to think , which trigonometric formula fits into the given equation so that the problem is solved. Then we have to use quadrant rules to write the general values of angles.