Prove that: cos(π/15) cos(2π/15) cos(4π/15) cos(7π/15) = 1/16

Question

Prove that:

cos\(\frac{\pi}{15}\) cos\(\frac{2\pi}{15}\) cos\(\frac{4\pi}{15}\) cos\(\frac{7\pi}{15}\) = \(\frac{1}{16}\)

Answer 1

LHS =  cos\(\frac{\pi}{15}\) cos\(\frac{2\pi}{15}\) cos\(\frac{4\pi}{15}\) cos\(\frac{7\pi}{15}\)

Multiply and divide by 2 sin\(\frac{\pi}{15},\) we get

Then the above equation becomes,

Multiply and divide by 2, we get

But 2sin A cos A = sin 2A

Then the above equation becomes,

But 2sin A cos B = sin (A + B) + sin(A – B), so the above equation becomes,