8Sin20°Sin40°Sin80° =√3
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Step-by-step explanation:
sin(20)sin(40) = (1/2)[cos(20) - cos(60)]
=> sin(20)sin(40)sin(80)
= (1/2)[cos(20) - cos(60)]sin(80)
= (1/2)[cos(20)sin(80) - cos(60)sin(80)]
= (1/2)[(1/2)(sin(100)+sin(60)) - (1/2)(sin(140)+sin(20))]
= (1/4)[sin(80) + sin(60) - sin(40) - sin(20)]
= (1/4)sin(60), since sin(80) - sin(40) - sin(20) = sin(60+20) - sin(60-20) - sin(20) = 2cos(60)sin(20) - sin(20) = 0
= (3^1/2)/8
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