Question

If $\tan \theta_{1}, \tan \theta_{2}, \tan \theta_{3}, \tan \theta_{4}$, are the roots of the equation $x^{4}-x^{3} \sin 2 \beta+x^{2} \cos 2 \beta-x \cos \beta-\sin \beta=0$ then $\tan \left(\theta_{1}+\theta_{2}+\theta_{3}+\theta_{4}\right)$ is

• A. $\sin \beta$
• B. $\cos \beta$
• C. $\tan \beta$
• D. $\cot \beta$

Answer 1

Answer: (D)

Given equation
$ x^{4}-x^{3} \sin (2 \beta)+x^{2} \cos (2 \beta)-x \cos \beta-\sin \beta=0$
$\therefore S_{1}=\sin (2 \beta), S_{2}=\cos 2 \beta, S_{3}=\cos \beta, S_{4}=-\sin \beta$
Now $\tan \left(\theta_{1}+\theta_{2}+\theta_{3}+\theta_{4}\right)=\frac{ s _{1}- s _{3}}{1- s _{2}+ s _{4}} $
$=\frac{\sin (2 \beta)-\cos \beta}{1-\cos 2 \beta-\sin \beta}$
$=\frac{2 \sin \beta \cos \beta-\cos \beta}{2 \sin ^{2} \beta-\sin \beta}=\cot \beta$