Question

$cos\frac{2\pi }{15}cos\frac{4\pi }{15}cos\frac{8\pi }{15}cos\frac{14\pi }{15}$

• A. $1/4$
• B. $1/8$
• C. $1/16$
• D. $1/32$

Answer 1

The correct option is C $1/16$

$\Rightarrow \frac{2\pi }{15}=\frac{2\times 180°}{15}=24°$

$\Rightarrow \frac{4\pi }{15}=48°$

$\Rightarrow \frac{8\pi }{15}=96°$

$\Rightarrow \frac{16\pi }{15}=192°$

$\therefore \cos 24°\cos 48°\cos 96°\cos 192°$

Now, multiply and divide by $16\sin 24°,$ we get

$=\frac{1}{16\sin 24°}\left[\left(2\sin 24°\cos 24°\right)2\cos 48°2\cos 96°2\cos 192°\right]$

$=\frac{1}{16\sin 24°}\left[\left(2\sin 48°\cos 48°\right)2\cos 96°2\cos 192°\right]$

$=\frac{1}{16\sin 24°}\left[\left(2\sin 96°\cos 96°\right)2\cos 192°\right]$

$=\frac{1}{16\sin 24°}\left[2\sin 192°2\cos 192°\right]$

$=\frac{1}{16\sin 24°}\times \sin 384°$

$=\frac{1}{16\sin 24°}\times \sin (360°+24°)$

$=\frac{1}{16\sin 24°}\times \sin 24°$

$=\frac{1}{16}$

Hence, the answer is $\frac{1}{16}.$