Question

sin pi by 5 into sin 2 pi by 5 into sin 45 by 5 into sin 3 pi by 5 is equals to 5 by 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.