Use the geometric series e^(ix) - (1/2)e^(3ix) + (1/4)e^(5ix) - ... to find the exact value sin1 -(1/2)sin3 + (1/4)sin5 - ...

S = e^ix - (1/2)e^3ix + (1/4)e^5ix - … is an infinite geometric series, equal to a/(1 - r). a = e^ix and r = (1/2)e^2ix thus S = e^ix/(1+(1/2)e^2ix = 2e^ix/(2+e^2ix). Rationalising the denominator: S = 2e^ix(2+e^-2ix)/(2+e^2ix)(2+e^-2ix) = (4e^ix + 2e^-ix)/(4 + 2(e^2ix + e^-2ix) + 1) = (4(cosx + isinx) + 2(cosx - isinx))/(5 + 2(2cosx)) = 6cosx/(5 + 4cos2x) + i(2sinx/(5 + 4cos2x)). We know that S = cosx + isinx - (1/2)cos3x -(1/2)isin3x+ (1/4)cos5x + (1/4)isin5x - … Hence Im(S) = sinx - (1/2)sin3x + (1/4)sin5x - … = 2sinx/(5+4cosx). Hence sin1 - (1/2)sin3 + (1/4)sin5 - … = 2sin1/(5 + 4cos2)