Question

Prove that:
(a) sin20°. sin 40°. sin60°. sin80° = 3 ÷ 16​

Answer 1

since,sin60=√ 3/2

= √ 3/2( sin20sin40sin80)

=√ 3/2( sin20sin80sin40)

=√ 3/4 [(2sin20sin40)sin80]

on applying [cos(A-B)-cos(A+B) = 2sinAsinB]

we get,

= √ 3/4 (cos20-cos60)sin80 [since,cos(-a)=cosa]

= √ 3/4(cos20sin80-cos60sin80)

= √ 3/8(2sin80cos20-sin80)

= √ 3/8(sin100+sin60-sin80)

= √ 3/8( √ 3/2+sin100-sin80 )

= √ 3/8( √ 3/2+sin(180-80)-sin80 )

= √ 3/8( √ 3/2+sin80-sin80 ) [since,sin(180-a)=sina]

= √ 3/8( √ 3/2)

= 3/16

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