Prove that : sin^2 (π/18) + sin^2 (π /9) + sin^2 (7π /18) + sin^2(4π/9) = 2

Question

Prove that :

sin² (\(\cfrac{\pi}{18}\)) + sin² (\(\cfrac{\pi}9\)) + sin² (\(\cfrac{7\pi}{18}\)) + sin²(\(\cfrac{4\pi}9\)) = 2

Answer 1

LHS = sin² (\(\cfrac{\pi}{18}\)) + sin² (\(\cfrac{\pi}9\)) + sin² (\(\cfrac{7\pi}{18}\)) + sin²(\(\cfrac{4\pi}9\))

We know that when n is odd, sin → cos.

 

We know that sin²x + cos²x = 1.

= 1 + 1 = 2

= RHS

Hence proved.