Question 132365
You can't verify the statement as you presented it.  That's because, in general,{{{2sin(x)cos^3(x)+2sin^2(x)cos(x) <> sin(2x)}}}


I suspect that you made a typing error when you presented the problem and what you meant was:


{{{2sin(x)cos^3(x)+red(2sin^3(x))cos(x) = sin(2x)}}}


In that case, factor out {{{2sin(x)cos(x)}}} on the left.


{{{(2sin(x)cos(x))(cos^2(x)+sin^2(x)) =sin(2x)}}}, but {{{cos^2(x)+sin^2(x)=1}}}, so {{{2sin(x)cos(x) =sin(2x)}}}, which is the standard form of the Double Angle Formula for the sine function.  If you need to prove that part, just Google 'Double Angle Formula' and you will find several proofs.