Question

The expression $\cos \frac{10\mathrm{\pi }}{13}+\cos \frac{8\mathrm{\pi }}{13}+\cos \frac{3\mathrm{\pi }}{13}+\cos \frac{5\mathrm{\pi }}{13}$ is

• A. $-1$
• B. $0$
• C. $1$
• D. None of these

Answer 1

The correct option is B

$0$

Calculate the given expression:

The given equation is $\cos \frac{10\mathrm{\pi }}{13}+\cos \frac{8\mathrm{\pi }}{13}+\cos \frac{3\mathrm{\pi }}{13}+\cos \frac{5\mathrm{\pi }}{13}$

$$\begin{gathered} =\cos \left(\mathrm{\pi }-\frac{3\mathrm{\pi }}{13}\right)+\cos \left(\mathrm{\pi }-\frac{5\mathrm{\pi }}{13}\right)+\cos \frac{3\mathrm{\pi }}{13}+\cos \frac{5\mathrm{\pi }}{13} \\ =-\cos \frac{3\mathrm{\pi }}{13}-\cos \frac{5\mathrm{\pi }}{13}+\cos \frac{3\mathrm{\pi }}{13}+\cos \frac{5\mathrm{\pi }}{13} \\ =0 \end{gathered}$$

Hence, the value of the given expression is $0$ and therefore option $\left(B\right)$ is correct.