Prove That cos20° cos40° cos80° = 1/8 -This is an Important Sum of Class 11 Trigonometry of Transformations of Sums and Products Chapter. This Problem is in The Books of SN Dey, RD Sharma, ML Aggarwal, NCERT, Etc.
Prove That cos20° cos40° cos80° = $\frac{1}{8}$
Solution:
cos20° cos40° cos80°
= $\frac{1}{2}$(2 cos20° cos40°) cos80°
= $\frac{1}{2}${cos (20°+40°) + cos(20°-40°)} cos80°
= $\frac{1}{2}$ {cos60° +cos(-20°)} cos80°
= $\frac{1}{2}$ (cos60° +cos20°) cos80°
= $\frac{1}{2}$ ($\frac{1}{2}$ +cos20°) cos80°
=$\frac{1}{4}$ cos80° + $\frac{1}{2}$cos20° cos80°
= $\frac{1}{4}$ cos80°+ $\frac{1}{4}$ (2 cos20° cos80°)
= $\frac{1}{4}$ cos80°+ $\frac{1}{4}$ {cos(20°+80°) + cos(20°-80°)}
= $\frac{1}{4}$cos80°+ $\frac{1}{4}$ {cos100° +cos(-60°)}
= $\frac{1}{4}$cos80°+$\frac{1}{4}$ (cos100° +cos60°)
= $\frac{1}{4}$cos80°+ $\frac{1}{4}$cos(180°-80°) + $\frac{1}{4}$ cos60°
= $\frac{1}{4}$cos80°+ $\frac{1}{4}$(-cos80°) + $\frac{1}{4}$ × $\frac{1}{2}$
= $\frac{1}{4}$ cos80° – $\frac{1}{4}$cos80° + $\frac{1}{8}$
= $\frac{1}{8}$ [Proved]