Question

Prove: $\cos \frac{\pi }{12}+\cos \frac{3\pi }{12}+\cos \frac{9\pi }{12}+\cos \frac{11\pi }{12}=0$

Answer 1

$\cos \frac{\pi }{12}+\cos \frac{3\pi }{12}+\cos \frac{9\pi }{12}+\cos \frac{11\pi }{12}$

$=2\cos \frac{(\frac{\pi }{12}+\frac{3\pi }{12})}{2}\cos \frac{(\frac{\pi }{12}-\frac{3\pi }{12})}{2}+2\cos \frac{(\frac{9\pi }{12}+\frac{11\pi }{12})}{2}\cos \frac{(\frac{9\pi }{12}-\frac{11\pi }{12})}{2}$

$=2\cos \frac{4\pi }{12\times 2}\cos \frac{2\pi }{12\times 2}+2\cos \frac{20\pi }{12\times 2}\cos \frac{2\pi }{12\times 2}$

$=2\cos \frac{\pi }{6}\cos \frac{\pi }{12}+2\cos \frac{5\pi }{5}\cos \frac{\pi }{12}$

$=2\cos \frac{\pi }{12}[\cos \frac{\pi }{6}+\cos \frac{5\pi }{6}]$

$=2\cos \frac{\pi }{12}[\cos \frac{\pi }{6}+\cos (\pi -\frac{\pi }{6})]$

$=2\cos \frac{\pi }{12}[\cos \frac{\pi }{6}-\cos \frac{\pi }{6}]=0$.