Half range cosine series

Half-range cosine series on (0, π): f(x) = a₀/2 + Σ aₙ cos nx, with a₀ = (2/π)∫₀^π f dx and aₙ = (2/π)∫₀^π f cos nx dx.

For f(x) = sin x:

a₀ = (2/π) ∫₀^π sin x dx = 4/π

For aₙ use 2 sin x cos nx = sin(n+1)x − sin(n−1)x:

a₁ = (2/π) ∫₀^π sin x cos x dx = (1/π) ∫₀^π sin 2x dx = 0
aₙ = (1/π) ∫₀^π [ sin(n+1)x − sin(n−1)x ] dx    (n ≥ 2)
   = (1/π) [ (1 − (−1)ⁿ⁺¹)/(n+1) − (1 − (−1)ⁿ⁻¹)/(n−1) ]

For odd n this is 0. For even n: aₙ = (1/π)[2/(n+1) − 2/(n−1)] = −4/(π(n² − 1)).

Answer (put n = 2m):

sin x = 2/π − (4/π) Σ_{m=1}^∞ cos 2mx / (4m² − 1)
      = 2/π − (4/π) [ cos 2x/3 + cos 4x/15 + cos 6x/35 + ... ],   0 ≤ x ≤ π

(Check at x = 0: 0 = 2/π − (4/π)(1/2), since Σ 1/(4m² − 1) = 1/2 ✓.)