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 $\cos \frac{10 \pi}{13}+\cos \frac{3 \pi}{13} \neq \cos \frac{13 \pi}{13}$