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प्रश्न
Prove the following :
sin 20° sin 40° sin 60° sin 80° = `3/16`
उत्तर
L.H.S. = sin 20°· sin 40°· sin 60°· sin 80°
= `sqrt(3)/2*sin20^circ* sin40^circ * sin80^circ .....[because sin60^circ = sqrt(3)/2]`
= `sqrt(3)/4(2 sin40^circ* sin20^circ)*sin80^circ`
= `sqrt(3)/4[cos(40^circ - 20^circ) - cos(40^circ + 20^circ)] xx sin80^circ`
= `sqrt(3)/4[cos20^circ - cos60^circ]*sin80^circ`
= `sqrt(3)/8[2 sin80^circ* cos20^circ - 2 cos 60^circ* sin80^circ]`
= `sqrt(3)/8[sin(80^circ + 20^circ) + sin(80^circ - 20^circ) - 2 xx 1/2* sin80^circ]`
= `sqrt(3)/8[sin100^circ + sin60^circ - sin80^circ]`
= `sqrt(3)/8[sin(180^circ - 80^circ) + sqrt(3)/2 - sin80^circ]`
= `sqrt(3)/8(sin80^circ + sqrt(3)/2 - sin80^circ)`
= `sqrt(3)/8 xx sqrt(3)/2`
= `3/16`
= R.H.S.
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