Question

The expression $2\cos \frac{\mathrm{\pi }}{13}\cos \frac{9\mathrm{\pi }}{13}+\cos \frac{3\mathrm{\pi }}{13}+\cos \frac{5\mathrm{\pi }}{13}$ is equal to

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

Answer 1

The correct option is B

$0$

Evaluate the given expression:

The given equation is

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

Using the formula $2\cos A\cos B=\cos \left(A+B\right)+\cos \left(A-B\right)$

$$\begin{gathered} \Rightarrow \cos \left(\frac{\mathrm{\pi }}{13}+\frac{9\mathrm{\pi }}{13}\right)+\cos \left(\frac{9\mathrm{\pi }}{13}-\frac{\mathrm{\pi }}{13}\right)+\cos \left(\frac{3\mathrm{\pi }}{13}\right)+\cos \left(\frac{5\mathrm{\pi }}{13}\right) \\ =\cos \left(\frac{10\mathrm{\pi }}{13}\right)+\cos \left(\frac{8\mathrm{\pi }}{13}\right)+\cos \left(\frac{3\mathrm{\pi }}{13}\right)+\cos \left(\frac{5\mathrm{\pi }}{13}\right) \\ =\left\{\cos \left(\frac{10\mathrm{\pi }}{13}\right)+\cos \left(\frac{3\mathrm{\pi }}{13}\right)\right\}+\left\{\cos \left(\frac{8\mathrm{\pi }}{13}\right)+\cos \left(\frac{5\mathrm{\pi }}{13}\right)\right\} \end{gathered}$$

Using the formula $\cos x+\cos y=2\cos \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right),$

$$\begin{gathered} =2\cos \left(\frac{{\displaystyle \frac{10\mathrm{\pi }}{13}}+\frac{3\mathrm{\pi }}{13}}{2}\right)\cos \left(\frac{{\displaystyle \frac{10\mathrm{\pi }}{13}-\frac{3\mathrm{\pi }}{13}}}{2}\right)+2\cos \left(\frac{{\displaystyle \frac{8\mathrm{\pi }}{13}+\frac{5\mathrm{\pi }}{13}}}{2}\right)\cos \left(\frac{{\displaystyle \frac{8\mathrm{\pi }}{13}-\frac{5\mathrm{\pi }}{13}}}{2}\right) \\ =2\cos \left(\frac{\mathrm{\pi }}{2}\right)\cos \left(\frac{7\mathrm{\pi }}{26}\right)+2\cos \left(\frac{\mathrm{\pi }}{2}\right)\cos \left(\frac{3\mathrm{\pi }}{26}\right) \\ =2\left(0\right)\cos \left(\frac{7\mathrm{\pi }}{26}\right)+2\left(0\right)\cos \left(\frac{3\mathrm{\pi }}{26}\right) \\ =0 \end{gathered}$$

Hence, the given expression is equal to $0$ and therefore the correct answer is option (B).