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Question
Show that sin 12° sin 48° sin 54° = `1/8`
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Solution
sin 12° . sin 48° . sin 54° = sin 48° . sin 12° . sin(90° – 36°)
= `1/2` [cos (48° – 12°) – cos (48° + 12°)] cos 36°
= `1/2` [cos 36° – cos 6o°] cos 36°
= `1/2 [cos 36^circ - 1/2] cos 36^circ`
= `1/2 [cos^2 36^circ - 1/2 cos 36^circ]`
= `1/2 [((sqrt(5) + 1)/4)^2 - 1/2((sqrt(5) + 1)/4)]`
= `1/2[(5 + 2sqrt(5) + 1)/16 - ((sqrt(5) + 1)/8)]`
= `1/16 [(5 + 2sqrt(5) + 1)/2 - (sqrt(5) + 1)]`
= `1/16 [(6 + 2sqrt(5) - 2sqrt(5) - 2)/2]`
= `1/16 xx 4/2`
= `1/8`
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