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प्रश्न
Prove the following :
cos 20° cos 40° cos 60° cos 80° = `1/16`
उत्तर
L.H.S. = cos 20°· cos 40° · cos 60° · cos 80°
`= 1/2 xx (2 cos 20^circ * cos 40^circ) xx 1/2 xx cos 80^circ ...[2 cos "A" * cos "B" = cos ("A + B") + cos ("A" * "B")]`
`= 1/2 [cos 60^circ + cos (- 20^circ)] xx 1/2 xx cos 80^circ`
`= 1/2 (cos 60^circ + cos 20^circ) xx 1/2 xx cos 80^circ`
`= 1/2 (1/2 + cos 20^circ) xx 1/2 cos 80^circ`
`= 1/8 cos 80^circ + 1/4 cos 20^circ * cos 80^circ`
`= 1/8 cos 80^circ + 1/4 xx 1/2 (2 cos 20^circ * cos 80^circ) ...[2 cos "A" + cos "B" = cos ("A + B") + cos ("A - B")]`
`= 1/8 cos 80^circ + 1/8 [cos 100^circ + cos (- 60)^circ]`
`= 1/8 cos 80^circ + 1/8 [cos (180^circ - 80^circ) + cos 60^circ]`
`= 1/8 cos 80^circ + 1/8 [- cos 80^circ + 1/2]`
`= 1/8 cos 80^circ - 1/8 cos 80^circ + 1/16`
`= cancel(1/8 cos 80^circ) - cancel(1/8 cos 80^circ) + 1/16`
`= 1/16`
= R.H.S.
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