If are the solutions of the equation (where no two solutions among these have equal tangents), then value of is

JEE MainMathematicsModerateTrigonometry

If $\alpha,\beta,\gamma,\delta$ are the solutions of the
equation $\tan\left( \theta + \frac{\pi}{4} \right) = 3tan3\theta$
(where no two solutions among these have equal tangents), then value of
$tan\alpha + tan\beta + tan\gamma + tan\delta$ is

Show answer and solution

\tan\left( \theta + \frac{\pi}{4} \right) = 3\tan\theta,\frac{1 + \tan\theta}{1 - \tan\theta} = 3\left( \frac{3\tan\theta - \tan^{3}\theta}{1 - 3\tan^{2}\theta} \right)

Let

tan\theta = t \Rightarrow \frac{1 + t}{1 - t} = 3\left( \frac{3t - t^{3}}{1 - 3t^{2}} \right)

\Rightarrow

3t^4 - 6t^2 + 8t -1= 0 so, t_1 + t_2 + t_3 + t_4 =
0

\therefore tan\alpha + tan\beta + tan\gamma + tan\delta = 0

Source: Grass-G / M-23 Mathematics Paper 31 Dec FST.docx

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