Q120 / 200 · Manipal (MET) 2012 · Mathematics
The value of $\cos \frac{2 \pi}{7}+\cos \frac{4 \pi}{7}+\cos \frac{6 \pi}{7}$ is
- A1
- B-1
- C$\frac{1}{2}$
- Correct answerD$-\frac{1}{2}$
Solution
$\therefore \cos \frac{2 \pi}{7}+\cos \frac{4 \pi}{7}+\cos \frac{6 \pi}{7}$
$\begin{gathered}=\frac{1}{2 \sin \frac{\pi}{7}} \times \\ {\left[2 \sin \frac{\pi}{7} \cos \frac{2 \pi}{7}+2 \sin \frac{\pi}{7} \cos \frac{4 \pi}{7}+2 \sin \frac{\pi}{7} \cos \frac{6 \pi}{7}\right]}\end{gathered}$
$\begin{gathered}=\frac{1}{2 \sin \frac{\pi}{7}} \times \\ {\left[\sin \frac{3 \pi}{7}-\sin \frac{\pi}{7}+\sin \frac{5 \pi}{7}-\sin \frac{3 \pi}{7}+\sin \pi-\sin \frac{5 \pi}{7}\right]} \\ =-\frac{1}{2}\end{gathered}$
Hence, the correct answer is option: 'D'.