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Prove that sin 12°. sin 48°. sin 54° = 1/8. |
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Answer» L.H.S=sin(12°)sin(48°)sin(54°) = [sin(12°)sin(48°)]sin(54°) = sin(54°) * [cos(48° - 12°) - cos(48° + 12°)]/2 = sin(54°) * [cos(36°) - cos(60°)]/2 = sin(54°) * [cos(36°) - 1/2]/2 = cos(36°) * [cos(36°)/2 - 1/4] = cos2(36°)/2 - cos(36°)/4. Since cos(36°) = (1 + √5)/4: cos2(36°)/2 - cos(36°)/4 = [(1 + √5)/4]2/2 - [(1 + √5)/4]/4 = (1 + √5)2/32 - (1 + √5)/16 = (6 + 2√5)/32 - (1 + √5)/16 = (6 + 2√5)/32 - (2 + 2√5)/32 = 4/32 = 1/8 R.H.S |
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