# Prove the following : cos 20° cos 40° cos 60° cos 80° = 116 - Mathematics and Statistics

Sum

Prove the following :

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

#### Solution

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.

Concept: Factorization Formulae - Formulae for Conversion of Product in to Sum Or Difference
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