prove that : sin20 sin40 sin60 sin80 = 3/16
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LHS =sin20°.sin40°.sin60°.sin80°
= 1/2{ 2sin20.sin40°}sin60.sin80°
= 1/2{ cos20° -cos60°} sin60°sin80°
= 1/4×√3/2 (2sin80.cos20°} -1/8{ 2sin60.sin80°}
=√3/8 { sin100° + sin60°} -√3/8 sin80°
=√3/8sin80° + √3/8 × √3/2 - √3/8sin80°
=3/16 = RHS
= 1/2{ 2sin20.sin40°}sin60.sin80°
= 1/2{ cos20° -cos60°} sin60°sin80°
= 1/4×√3/2 (2sin80.cos20°} -1/8{ 2sin60.sin80°}
=√3/8 { sin100° + sin60°} -√3/8 sin80°
=√3/8sin80° + √3/8 × √3/2 - √3/8sin80°
=3/16 = RHS
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