Question

The value of $\cos {20}^o\cos {40}^o\cos {60}^o\cos {80}^o$ is:

• A. $\frac{1}{2}$
• B. $\frac{1}{4}$
• C. $\frac{1}{8}$
• D. $\frac{1}{16}$

Answer 1

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

$\sin 2\theta =2\cos \theta \sin \theta$

Therefore

$\cos \theta =\frac{\sin 2\theta }{2\sin \theta }$ ... $\left(i\right)$

$\cos {20}^o\cos {40}^o\cos {60}^o\cos {80}^o$

$=\cos {20}^o\cos {40}^o\left(\frac{1}{2}\right)\cos {80}^o$

$=\frac{1}{2}\left(\cos {20}^o\cos {40}^o\cos {80}^o\right)$

$=\frac{1}{2}\left(\frac{\sin {40}^o}{2\sin {20}^o}\frac{\sin {80}^o}{2\sin {40}^o}\frac{\sin {160}^o}{2\sin {80}^o}\right)$ ... From $\left(i\right)$

$=\frac{1}{8}\left(\frac{\sin {160}^o}{2\sin {20}^o}\right)$

$=\frac{1}{8}\left(\frac{\sin ({180}^o-{20}^o)}{2\sin {20}^o}\right)$

$=\frac{1}{8}\left(\frac{\sin \left({20}^o\right)}{2\sin {20}^o}\right)$

$=\frac{1}{16}$