Question

Prove that : sin pi/5 * sin 2pi/5 * sin 3pi/5 * sin 4pi/5 = 5/16

Answer 1

If A + B = π, then
⇒ A = π - B ⇒ sin A = sin(π-B) = sin A = sin B
∴ π/5 + 4π/5 = π ⇒ sin π/5 = sin 4π/5 and 2π/5 + 3π/5 = π ⇒ sin 2π/5 = sin 3π/5
∴ L.H.S. = sin π/5 sin 2π/5 sin 3π/5 sin 4π/5
= L.H.S. = sin π/5 sin 2π/5 sin 2π/5 sin π/5
⇒ L.H.S. = (sin π/5 sin 2π/5)² = (sin 36° sin 72°)² = (sin 36° sin 18°)²
⇒ L.H.S.= {(√10-2√5/4) × (√10+2√5/4)}² = (10-2√5/16) × 10+2√5/16
⇒ L.H.S = 100-20/256 = 80/256 = 5/16 = R.H.S.
L.H.S. = R.H.S. hence proved.

Answer 2

Answer:

5/16

Step-by-step explanation

LHS---sin pi/5 *sin 2 pi/5 * sin 3 pi/5* sin 4 pi/5

=sin pi/5* sin 2 pi/5* sin(pi-2 pi/5) * sin(pi-pi/5)

=sin pi/5* sin 2 pi/5* sin pi/5*  sin 2 pi/5

=sin square pi/5* sin square 2 pi/5

=sin square 2 pi/5 * 2sin pi/5 cos pi/5

=4 sin to the power of 4 pi/5.cos square pi/5

(sin square pi/5=10-2 root 5/16,,,,cos pi/5= root 5+1/4)

= 4.100+20-40 root 5/256 multiplied 6+2 root 5/16

= 40(3- root 5).2(3+ root 5)/256.4

=5×4/64=5/16

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