Prove that: {sin}^(2) \: (7\pi)/(8) + \: {sin}^(2) \: (5\pi)/(8) = 1 ​

Question

Prove that:


`{sin}^(2) \: (7\pi)/(8) + \: {sin}^(2) \: (5\pi)/(8) = 1`
​

Answer 1

Recall that

sin²(x) = (1 - cos(2x))/2

so that

sin²(7π/8) = (1 - cos(7π/4))/2

sin²(5π/8) = (1 - cos(5π/4))/2

Then

sin²(7π/8) + sin²(5π/8) = (1 - cos(7π/4) + 1 - cos(5π/4))/2

… = 1 - 1/2 (cos(7π/4) + cos(5π/4))

… = 1 - 1/2 (1/√2 - 1/√2)

… = 1