​ Prove that: cos 20° cos 40° cos 80° = ​1/8

Question

Prove that: cos 20° cos 40° cos 80° = \(\frac{1}{8}\)

Answer 1

L.H.S = cos 20° cos 40° cos 80° 

= \(\frac{(2sin20°cos20°)cos40°cos80°}{2sin20°}\)

= \(\frac{sin(2\times30°)cos40°cos80°}{2sin20°}\)

[∵ 2 sin A cos A = sin 2A] 

= \(\frac{sin40°cos40°cos80°}{2sin20°}\) 

= \(\frac{(2sin40°cos40°)cos80°}{2\times sin20°}\) 

= \(\frac{sin(2\times 40°)cos80°}{4\times sin20°}\) 

= \(\frac{sin80°cos80°}{4sin20°}\) 

= \(\frac{2sin80°cos80°}{8sin20°}\) 

= \(\frac{sin(2\times 80°)}{8sin20°}\)

= \(\frac{sin160°}{8sin20°}\)

= \(\frac{sin(180°-20°)}{8sin20°}\)

= \(\frac{sin20°}{8sin20°}\) 

= \(\frac{1}{8}\)