Prove that cos ⁴π\8+cos⁴3π/8+cos⁴5π/8+cos⁴7π/8=3/2
Using the identity,
cos 2α = 1 – 2sin^2(α) = 2cos^2(α) – 1
then,
sin^2(α) = (1/2)(1 – cos 2α ), cos^2(α) = (1/2)(1 + cos 2α )
cos ⁴π/8
cos^2(π/8) = (1/2)(1 + cos π/4) = (1.2)(1 + 1/√2) = (√2 + 1)/(2√2)
cos⁴3π/8
cos 3π/8 = cos(π/2 – π/8) = cos(π/2).cos(π/8) + sin(π/2).sin(π/8)
cos 3π/8 = sin(π/8)
cos^2(3π/8) = sin^2(π/8) = (1/2)(1 – cos π/4) = (1/2)(1 – 1/√2) = -(√2 – 1)/(2√2)
cos⁴5π/8
cos 5π/8 = cos(π/2 + π/8) = cos(π/2).cos(π/8) – sin(π/2).sin(π/8)
cos 5π/8 = -sin(π/8)
cos^2(5π/8) = sin^2(π/8) = (1/2)(1 – cos π/4) = (1/2)(1 – 1/√2) = -(√2 – 1)/(2√2)
cos⁴7π/8
cos 7π/8 = cos(π – π/8) = cos(π).cos(π/8) + sin(π).sin(π/8) = -cos(π/8)
cos^2(7π/8) = cos^2(π/8) = (1/2)(1 + cos π/4) = (1.2)(1 + 1/√2) = (√2 + 1)/(2√2)
Using the above results in the initial identity to be proved,
cos ⁴π\8+cos⁴3π/8+cos⁴5π/8+cos⁴7π/8=3/2
(√2 + 1)^2/(4*2) + (√2 – 1)^2/(4*2) + (√2 – 1)^2/(4*2) + (√2 + 1)^2/(4*2) = 3/2
{(√2 + 1)^2 + (√2 – 1)^2 + (√2 – 1)^2 + (√2 + 1)^2}/(4*2) = 3/2
{(√2 + 1)^2 + (√2 – 1)^2}/(4) = 3/2
{2 + 2√2 + 1 + 2 – 2√2 + 1}/4 = 3/2
{3 + 3)/4 = 3/2
3/2 = 3/2