Question

Prove that

Cos 20° Cos 40° Cos 60° Cos 80° = 1/16​

Answer 1

Answer:

Cos20°cos40°cos60°cos80°

=(1/2)(2cos20°cos40°)(1/2)cos80° [∵,cos60°=1/2]

=(1/4)[cos(20°+40°)+cos(20°-40°)]cos80°

=(1/4)(cos60°+cos20°)cos80°

=(1/4)(cos60°cos80°+cos20°cos80°)

=(1/4)(1/2)cos80°+(1/4)cos20°cos80°

=(1/8)cos80°+(1/4)(1/2)(2cos20°cos80°)

=(1/8)cos80°+(1/8)[cos(20°+80°)+cos(20°-80°)]

=(1/8)cos80°+(1/8)(cos100°+cos60°)

=(1/8)cos80°+(1/8)cos100°+(1/8)cos60°

=(1/8)(cos80°+cos100°)+(1/8)×(1/2)

=(1/8)[{2cos(80°+100°)/2}{cos(80°-100°)/2}]+(1/16)

=(1/8)(2cos90°cos10°)+(1/16)

=0+(1/16) [cos90°=0]

=1/16 (proved)