Question

Prove the following :

cos 20° cos 40° cos 60° cos 80° = `1/16`

Answer 1

L.H.S. = cos 20°. cos 40°. cos 60°. cos 80°

= `cos20^circ. cos40^circ. 1/2. cos80^circ`

= `1/(2 xx 2)(2cos 40^circ. cos20^circ). cos80^circ`

= `1/4[cos(40^circ + 20^circ) + cos(40^circ - 20^circ)]. cos80^circ`

= `1/4(cos60^circ +  cos20^circ)cos80^circ`

= `1/4cos60^circ. cos80^circ + 1/4 cos20^circ. cos80^circ`

= `1/4(1/2)cos 80^circ + 1/(2 xx 4)(2cos80^circ cos20^circ)`

= `1/8cos80^circ + 1/8[cos(80^circ + 20^circ) + cos(80^circ - 20^circ)]`

= `1/8cos80^circ + 1/8(cos100^circ +  cos60^circ)`

= `1/8cos80^circ + 1/8cos100^circ + 1/8cos60^circ`

= `1/8cos80^circ + 1/8cos(180^circ - 80^circ) + 1/8 xx 1/2`

= `1/8cos80^circ - 1/8cos80^circ + 1/16`   ...[∵ cos(180° – θ) = – cos θ]

= `1/16`

= R.H.S.