Prove that: sin(π/5) sin(2π/5) sin(3π/5) sin(4π/5) = 5/16

Question

Prove that:

sin\(\frac{\pi}{5}\) sin\(\frac{2\pi}{5}\) sin\(\frac{3\pi}{5}\) sin\(\frac{4\pi}{5}\) = \(\frac{5}{16}\)

Answer 1

LHS =  sin\(\frac{\pi}{5}\) sin\(\frac{2\pi}{5}\) sin\(\frac{3\pi}{5}\) sin\(\frac{4\pi}{5}\)

This can be rewritten as,

But sin(π - θ) = sin θ so the above equation becomes,

Then the above equation becomes,

Hence the above equation becomes,