Question

Prove that: (sin 3 x + sin x ) sin x + (cos 3 x – cos x ) cos x = 0

Answer 1

Consider the left hand side (LHS),

LHS=( sin3x+sinx )sinx+( cos3x−cosx )cosx =sin3xsinx+ sin 2 x+cos3xcosx− cos 2 x =cos3xcosx+sin3xsinx−( cos 2 x− sin 2 x ) =cos( 3x−x )−cos2x         ( ∵cos( A+B )=cosAcosB+sinAsinB )

Further simplify,

=cos2x−cos2x =0

Right hand side (RHS)

=0

Hence, LHS = RHS