Trig Identities

sin 4x   = (4sinxcosx) (2cos^2 (2x) - 1 )

4sinxcosx  =  2*2sinxcosx = 2sin2x

2cos^2(2x) - 1 =   cos(4x)

But.....as the graph below shows,

sin (4x)   is not equal to   2sin(2x)cos(4x)

So....this is not an identity.....

Can you prove sin4x=(4sinxcosx)(2cos^2x-1)?

\(\begin{array}{|rcll|} \hline \sin(4x) &=& [4\sin(x)\cos(x)][2\cos^2(x)-1] \\ && \cos(2x) = \cos^2(x) - \sin^2(x) = \cos^2(x) - [1- \cos^2(x)] = 2\cos^2(x)-1 \\ \sin(4x) &=& [4\sin(x)\cos(x)][\cos(2x)] \\ && \sin(2x) = 2\cdot \sin(x)\cos(x) \\ \sin(4x) &=& 2\sin(2x)\cos(2x) \\ && \sin(4x)=\sin(2x+2x) = \sin(2x)\cos(2x) + \cos(2x)\sin(2x) = 2\sin(2x)\cos(2x) \\ \sin(4x) &=& \sin(4x) \ \checkmark \\ \hline \end{array}\)