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Question: If \(\tan \theta + \tan 4\theta + \tan 7\theta = \tan \theta \tan 4\theta \tan 7\theta \) then the g...

If tan⁡θ+tan⁡4θ+tan⁡7θ=tan⁡θtan⁡4θtan⁡7θ\tan \theta + \tan 4\theta + \tan 7\theta = \tan \theta \tan 4\theta \tan 7\theta then the general solution is :-
A.θ=nπ4 B.θ=nπ12 C.θ=nπ6 D. None of these  {\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}} \\\

Explanation

Solution

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

The given equation is
tan⁡θ+tan⁡4θ+tan⁡7θ=tan⁡θtan⁡4θtan⁡7θ\tan \theta + \tan 4\theta + \tan 7\theta = \tan \theta \tan 4\theta \tan 7\theta
After transposing we get,
tan⁡θ+tan⁡4θ+tan⁡7θ−tan⁡θtan⁡4θtan⁡7θ=0 ............(i)\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)=tan⁡a+tan⁡b+tan⁡c−tan⁡atan⁡btan⁡c1−tan⁡atan⁡b−tan⁡atan⁡c−tan⁡btan⁡c ...........(ii)\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⁡(θ+4θ+7θ)=tan⁡θ+tan⁡4θ+tan⁡7θ−tan⁡θtan⁡4θtan⁡7θ1−tan⁡θtan⁡4θ−tan⁡θtan⁡7θ−tan⁡7θtan⁡4θ = tan⁡(12θ) ...........(iii)\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,
tan(12θ)=0 12θ=nπ θ = nπ12  {\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.