Question

Assertion :The number of solutions of the equation $\tan \theta +\tan 2\theta +\tan 3\theta =\tan \theta \tan 2\theta \tan 3\theta$ is two, $\forall 0<\theta <\pi$ Reason: $\tan 6\theta$ is not defined at $\theta =\frac{(2n+1)\pi }{12},\forall nϵN$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion false but Reason is true

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

$\tan \theta +\tan 2\theta +\tan 3\theta =\tan \theta \tan 2\theta \tan 3\theta$

$\Rightarrow \tan \theta +\tan 2\theta =-\tan 3\theta (1-\tan \theta +\tan 2\theta )$

$\Rightarrow \frac{\tan \theta +\tan 2\theta }{1-\tan \theta \tan 2\theta }=-\tan 3\theta$

$\Rightarrow \tan 3\theta =\tan (-3\theta )$

$\Rightarrow 3\theta =n\pi -3\theta$

$\Rightarrow 6\theta =n\pi \forall n\in Iorn=1,2,3,4,5$

$\Rightarrow \theta =\frac{n\pi }{6}$

As $0<\theta <\pi$

$\therefore \theta =\frac{\pi }{6},\frac{\pi }{3},\frac{\pi }{2},\frac{2\pi }{3},\frac{5\pi }{6}$

However $\tan \theta$ &$\tan 3\theta$ are not defined at $\frac{\pi }{2},\frac{\pi }{6},\frac{5\pi }{6}$ respectively, so,Reason (R) is correct but not the proper

explanation of Assertion (A) & $\frac{\pi }{3},\frac{2\pi }{3}$ are the only two solutions of equations.