Question

Consider the following statements
Statement I $2\cos \frac{\pi }{3}\cos \frac{9\pi }{13}+\cos \frac{3\pi }{13}+\cos \frac{5\pi }{13}=0$
Statement II $\cos \frac{10\pi }{13}+\cos \frac{3\pi }{13}=0$ and $\cos \frac{8\pi }{13}+\cos \frac{5\pi }{13}=0$
Choose the correct option.

• A. Statement I is true; Statement II is true; Statement II is a correct explanation for Statement I
• B. Statement I is true; Statement II is true; Statement II is not a correct explanation for Statement I
• C. Statement I is true; Statement II is false
• D. Statement I is false; Statement II is true

Answer 1

Answer: (A)

$2\cos \frac{\pi }{3}\cos \frac{9\pi }{13}+\cos \frac{3\pi }{13}+\cos \frac{5\pi }{13}$
$=\cos \left(\frac{9\pi }{13}+\frac{\pi }{13}\right)+\cos \left(\frac{9\pi }{13}-\frac{\pi }{13}\right)+\cos \frac{5\pi }{13}+\cos \frac{3\pi }{13}$
(by formulae)
$=\cos \frac{10\pi }{13}+\cos \frac{8\pi }{13}+\cos \frac{5\pi }{13}+\cos \frac{3\pi }{13}$
$=\left(\cos \frac{10\pi }{13}+\cos \frac{3\pi }{13}\right)+\left(\cos \frac{8\pi }{13}+\cos \frac{5\pi }{13}\right)$
$=2\cos \frac{\frac{10\pi }{13}+\frac{3\pi }{13}}{2}\cos \frac{\frac{10\pi }{13}-\frac{3\pi }{13}}{2}$
$+2\cos \frac{\frac{8\pi }{13}+\frac{5\pi }{13}}{2}\cos \frac{\frac{8\pi }{13}-\frac{5\pi }{13}}{2}$
(by formulae)
$=2\cos \frac{\pi }{2}\cos \frac{7\pi }{26}+2\cos \frac{\pi }{2}\cos \frac{3\pi }{26}=0+0=0$
$\left(\because \cos \frac{\pi }{2}=0\right)$
Note Remember that $\cos \frac{10\pi }{13}+\cos \frac{3\pi }{13}\ne \cos \frac{13\pi }{13}$