Prove that :sin5x - 2sin3x +sinx0055: - 005 2=tanx

1.	Prove that :sin5x - 2sin3x +sinx0055: - 005 2=tanx
Answer» We use sin A + sin B = 2 sin (A+B)/2. cos (A-B)/2cos A - cos B = - 2 sin (A+B)/2 sin(A-B)/2 sin5x-2sin3x+sinx)/(cos5x-cosx) = [{2sin(5x+x)/2cos(5x-x)/2}-2sin3x]/{2sin(5x+x)/2sin(x-5x)/2} = (2sin3xcos2x-2sin3x)/{2sin3xsin(-2x)} = {2sin3x(cos2x-1)}/{-2sin3xsin2x} = -(cos2x-1)/sin2x = (1-cos2x)/sin2x = 2sin^2 x/2sinxcosx = sinx/cosx = tanx (Proved)

Answer»

We use sin A + sin B = 2 sin (A+B)/2. cos (A-B)/2cos A - cos B = - 2 sin (A+B)/2 sin(A-B)/2

sin5x-2sin3x+sinx)/(cos5x-cosx)

= [{2sin(5x+x)/2cos(5x-x)/2}-2sin3x]/{2sin(5x+x)/2sin(x-5x)/2}

= (2sin3xcos2x-2sin3x)/{2sin3xsin(-2x)}

= {2sin3x(cos2x-1)}/{-2sin3xsin2x}

= -(cos2x-1)/sin2x

= (1-cos2x)/sin2x

= 2sin^2 x/2sinxcosx

= sinx/cosx

= tanx (Proved)

Discussion