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Find the value of sinpi/14sin3pi/14sin5pi/14

Sir 17th question please Find the value of sinpi/14sin3pi/14sin5pi/14
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@Gopal dixit

E =sin(π/14) sin(3π/14) sin(5π/14)

Put : x = π/14.

Then : 14x = π.

∴ E = sin x. sin 3x, sin 5x.

∴ E = sin (π/14). sin (3π/14). sin (5π/14) ......... (1)

Now,

sin π/14 = sin (π/2 - 6π/14) = cos 6π/14 = cos (π - 8π/14) = - cos 8π/14 = - cos 8x

Similarly, sin 3π/14 = cos 4π/14 = cos 4x and

sin 5π/14 = cos 2π/14 = cos 2x.

∴ from (1),

E = - cos 8x. cos 4x. cos 2x

Multiply and divide by 2 sin2x,

=  -1/ (2 sin2x ). [( 2 sin 2x. cos 2x ). cos 4x. cos 8x]

= (-1/ 4 sin 2x ). ( 2 sin 4x. cos 4x ). cos 8x

= (-1/ 8 sin 2x ). ( 2 sin 8x. cos 8x )

= (-1/ 8 sin 2x ). sin 16x

= (-1/8). ( 1/ (sin π/7)). sin 8π/7

= (-1/8). ( 1/ (sin π/7)). sin ( π + π/7 )

= - (1/8). ( 1/ sin π/7 ). ( - sin π/7 )

= 1/8.

∴ E = 1/8 

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