Question

how to prove that sin 20 sin 40 sin 60 sin 80 equal to 3 by 16

Answer 1

We know
,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 Ans

It's helpful for you ☺

Answer 2

DEAR STUDENT,

Kindly check the attached papers for a detailed solution along with step by step elaboration to reach our final solution that is our final desired or required answer to this query that is completely proved.

$\boxed{\bf{\underline{L.H.S. = R.H.S.}}}$

Which is the required proof or solution process for this type of query.

Hope this helps you and solves your doubts for applying trigonometric identifies to reach a proper solution or a proof!!!!!