Question

If $\alpha ,\beta ,\gamma ,\delta$ are the four solutions of the equation $\tan (\theta +\frac{\pi }{4})=3\tan 3\theta$ . No two of which have equal tangents, then the value of $\tan \alpha +\tan \beta +\tan \delta +\tan \gamma$

• A. $1$
• B. $0$
• C. $-1$
• D. $4$

Answer 1

The correct option is B $0$

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

$\Rightarrow 3\cdot {\tan }^4\theta +6\cdot {\tan }^2\theta -8.\tan \theta +1=0$

Since $\alpha ,\beta ,\gamma$ and $\delta$ are the roots

Sum of roots$=-\frac{coefficientof{\tan }^3\theta }{coefficientof{\tan }^4\theta }$

Therefore, $\tan \alpha +\tan \beta +\tan \gamma +\tan \delta =0=coefficientof{x}^3$