Question

If $\theta =\frac{2\pi }{7}$, then the value of tan $\theta$ tan $2\theta +tan2\theta$ tan $4\theta +tan4\theta$ tan $\theta$ is

• A. 8
• B. -8
• C. 7
• D. -7

Answer 1

The correct option is D

-7

Put $\theta =A,2\theta =B,4\theta =c$
$\therefore A+B+C=7\theta =2\pi$
$\therefore \sum tanAtanB=\frac{\sum sinAsinBcosC}{cosAcosBcosC}$
But cos (A+B+C)=cos A cos B cos $C-\sum$ sin A sin B cos C $\therefore \sum$ sin A sin B cos C=cos A cos B cos C -cos $2\pi$
$\therefore \sum tanAtanB=\frac{cosAcosBcosC-1}{cosAcosBcosC}=1-\frac{1}{cosAcosBcosC}$
$=1-\frac{1}{cos\frac{2\pi }{7}cos\frac{4\pi }{7}cos\frac{8\pi }{7}}=1-\frac{1}{\frac{sin{2}^3\left(\frac{2\pi }{7}\right)}{2^{3}sin\frac{2\pi }{7}}}$
=1-8=-7