Sum of cosines

Show that:

$$\cos \frac{\pi}{5} + \cos \frac{3\pi}{5}= \frac{1}{2}$$

Solution



Making use of the formula $\displaystyle \cos A + \cos B = 2\cos \frac{A+B}{2} \cos \frac{A-B}{2}$ successively we have:

$$\begin{aligned}
\cos \frac{\pi}{5}+\cos \frac{3\pi}{5} &=2\cos \frac{2\pi}{5}\cos \frac{\pi}{5} \\
 &=\frac{2\cos \frac{\pi}{5}\sin \frac{\pi}{5}\cos \frac{2\pi}{5}}{\sin \frac{\pi}{5}} \\
 &= \frac{\sin \frac{2\pi}{5}\cos \frac{2\pi}{5}}{\sin \frac{\pi}{5}}\\
 &= \frac{2\sin \frac{2\pi}{5}\cos \frac{2\pi}{5}}{2\sin \frac{\pi}{5}}\\
 &= \frac{\sin \frac{4\pi}{5}}{2\sin \frac{\pi}{5}}=\frac{\sin \left ( \pi-\frac{\pi}{5} \right )}{2\sin \frac{\pi}{5}}=\frac{\sin \frac{\pi}{5}}{2\sin \frac{\pi}{5}}=\frac{1}{2}
\end{aligned}$$