Question

Value of :

$2\cos \frac{\pi }{13}\cos \frac{9\pi }{13}+\cos \frac{3\pi }{13}+\cos \frac{5\pi }{13}$

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

Answer: (A)

$0$

given, $2\cos \frac{\pi }{13}\cos \frac{9\pi }{13}+\cos \frac{3\pi }{13}+\cos \frac{5\pi }{13}$

$=\cos (\frac{9\pi }{13}+\frac{\pi }{13})+\cos (\frac{9\pi }{13}-\frac{\pi }{13})+\cos \frac{3\pi }{13}+\cos \frac{5\pi }{13}$

$=\cos \frac{10\pi }{13}+\cos \frac{8\pi }{13}+\cos \frac{3\pi }{13}+\cos \frac{5\pi }{13}$

$=\cos (\pi -\frac{3\pi }{13})+\cos (\pi -\frac{5\pi }{13})+\cos \frac{3\pi }{13}+\cos \frac{5\pi }{13}$

$-\cos \frac{3\pi }{13}-\cos \frac{5\pi }{13}+\cos \frac{3\pi }{13}+\cos \frac{5\pi }{13}=0$