Question

The value of the expression $\cos \frac{\pi }{15}\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\sin \frac{\pi }{30}$ is

• A. $-\frac{1}{32}$
• B. $\frac{1}{32}$
• C. $-\frac{1}{16}$
• D. $\frac{1}{16}$

Answer 1

The correct option is D $\frac{1}{16}$

$\cos \frac{\pi }{15}\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\sin \frac{\pi }{30}$
$\Rightarrow \cos \frac{\pi }{15}\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\cos (\frac{\pi }{2}-\frac{\pi }{30})$
$\Rightarrow \cos \frac{\pi }{15}\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\cos \frac{7\pi }{15}$
$\Rightarrow \cos \frac{\pi }{15}\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\cos (\pi -\frac{8\pi }{15})$
$\Rightarrow -\cos \frac{\pi }{15}\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\cos \frac{8\pi }{15}$
$\Rightarrow -\frac{\sin (2^4.\frac{\pi }{15})}{2^4\sin \frac{\pi }{15}}=-\frac{\sin (\pi +\frac{\pi }{15})}{2^4\sin \frac{\pi }{15}}=\frac{1}{16}$