# 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}} \\

$

Last updated date: 25th Mar 2023

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Answer

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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.

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.

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