How does one obtain cos2x = cos^2(x) - sin^2(x) by differentiating the identity sin2x = 2sinxcosx

Answer 1

taking derivative of
sin2x = 2sinxcosx with respect to x

2cos2x = (2sinx)(-sinx) + cosx(2cosx)
cos2x = cos^2x - sin^2x

and there is your result.