Prove that, cos40° cos100° cos160°= $\frac{1}{8}$

Prove that, cos40° cos100° cos160°= $\frac{1}{8}$

Solution:

cos40° cos100° cos160°

= $\frac{1}{2}$ (2 cos100° cos40°) cos160°

= $\frac{1}{2}$  (cos 140° +cos 60°) cos160°

=$\frac{1}{2}$ $\left(\cos 140 + \frac{1}{2}\right)$ cos160°  

= $\frac{1}{2}$ cos140° cos160° + $\frac{1}{4}$  cos160°

=$\frac{1}{4}$ (2 cos160° cos140°) +$\frac{1}{4}$   cos160°

=$\frac{1}{4}${cos(160°+140°)+cos(160°-140°)} +$\frac{1}{4}$ cos160°

=  $\frac{1}{4}$  (cos300° +cos20°)+$\frac{1}{4}$   cos160°

= $\frac{1}{4}$ {cos(360°-60°)+cos20°}+$\frac{1}{4}$ cos(180°-20°)

= $\frac{1}{4}$ (cos60°+cos20°)-$\frac{1}{4}$ cos20°

= $\frac{1}{4}$ $\left(\frac{1}{2} + cos20°\right)$ – $\frac{1}{4}$ cos20°

=  $\frac{1}{8}$  + $\frac{1}{4}$ cos20° – $\frac{1}{4}$  cos20°

= $\frac{1}{8}$   [Proved]