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Question
Prove that:
cos 20° cos 40° cos 80° = `1/8`
Solution
cos 20° cos 40° cos 80°
= `((2 sin 20^circ)/(2 sin 20^circ))` cos 20° cos 40° cos 80°
[multiply and divide by 2 sin 20°]
`= ((2 sin 20^circ cos 20^circ) cos 40^circ cos 80^circ)/(2 sin 20^circ)`
`= (sin (2 xx 20^circ) cos 40^circ cos 80^circ)/(2 sin 20^circ)`
= `(sin 40^circ cos 40^circ cos 80^circ)/(2 sin 20^circ)`
(Multiply and divide by 2)
`= 1/2 xx ((2 sin 40^circ cos 40^circ))/(2 sin 20^circ) cos 80^circ`
`= 1/2 xx ((sin 2 xx 40^circ)cos 80^circ)/(2 sin 20^circ)`
`= 1/2 xx (sin 80^circ cos 80^circ)/(2 sin 20^circ)`
`= 1/2 xx 1/2 ((2 sin 80^circ cos 80^circ))/(2 sin 20^circ)`
`= 1/8 xx (sin 160^circ)/(sin 20^circ)`
`= 1/8 xx sin (180^circ - 20^circ)/(sin 20^circ)`
`= 1/8 xx sin 20^circ/sin 20^circ` ...[∵ sin(180° – θ) = sin θ]
`= 1/8 xx 1 = 1/8`
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