Numerator = (sin7x + sinx) − (sin5x + sin3x)
= 2sin4x⋅cos3x − 2sin4x⋅cosx (using C.D. formula)
= 2sin4x (cos3x − cosx)
Denominator = (cos 3x − cos5x) − (cosx − cos7x)
= 2sin4x sinx − 2sin4x sin3x
= 2sin4x (sinx − sin3x
Therefore, the given expression is
cos 3x - cos x/sin x - sin 3x = 2 sin 2x sin x/2cos 2x sin x = tan 2x
