Use the power reducing identities to rewrite each expression

Use the power reducing identities to rewrite each expression to one that contains a single trigonomic function of power 1.

4sinxcosx(2cos^2x-1)

Solution

y = 4sin x cos x [2cos²2x - 1]

=> y = 2 (2sin x cos x [2cos²2x - 1] )

Now sin 2x = 2 sin x cos x

therefore  y = 2 (sin2x [2cos²2x - 1])

Also  2cos²2x - 1 = cos 2 x

therefore  y = 2sin 2x cos 2x

and since sin 2x = 2 sin x cos x, therefore 2 sin 2x cos 2x = sin 4x

therefore y = sin 4x.

hence 4 sin x cos x [2cos²2x - 1] = sin 4x.