Question

If $7\left(\theta \right)=(2n+1)\pi$, when $n=0,1,2,3,4,5,6,$ then on the basis of given information, answer the given question.

The value of $\sec \frac{\pi }{7}+\sec \frac{3\pi }{7}+\sec \frac{5\pi }{7}$ is,

• A. $4$
• B. $-4$
• C. $3$
• D. $-3$

Answer 1

The correct option is A $4$

Put, $x=\frac{1}{y}$ in (ii) i.e., $(y=\sec \theta )$

then $\sec \frac{\pi }{7},\sec \frac{3\pi }{7},\sec \frac{5\pi }{7}$ are roots of a $\frac{8}{y^3}-\frac{4}{y^2}-\frac{4}{y}+1=0$

$\Rightarrow y^3-4y^2-4y+8=0$

$\therefore \sec \frac{\pi }{7}+\sec \frac{3\pi }{7}+\sec \frac{5\pi }{7}$ is $4......\left(iii\right)$

Option A