Prove that:

(sin(5x) - 2·sin(3x) + sin(x)) / (cos(5x) - cos(x)) = tan(x)

L.H.S. = (sin(5x) - 2 × sin(3x) + sin(x)) / (cos(5x) - cos(x))

= (sin(5x) + sin(x) - 2sin(3x)) / (cos(5x) - cos(x))

= (2sin(3x) × cos(2x) - 2sin(3x)) / (-2sin(3x) × sin(2x))

= -(sin(3x)(cos(2x) - 1)) / (sin(3x)sin(2x))

= (1 - cos(2x)) / sin(2x)

= (2sin²(x)) / (2sin(x)cos(x))

= tan(x) = R.H.S.