The answer is ##cos^2(x)-sin^2(x)+2cos(x)sin(x)=cos(2x)+sin(2x)##

First, use the to say

##d/dx(sin(x)(sin(x)+cos(x)))=cos(x)(sin(x)+cos(x))+sin(x)(cos(x)-sin(x))##

Next, expand this out to write

##d/dx(sin(x)(sin(x)+cos(x)))=cos^2(x)-sin^2(x)+2cos(x)sin(x)##

Finally, use the double-angle formulas ##cos^2(x)-sin^2(x)=cos(2x)## and ##2cos(x)sin(x)=sin(2x)## to write

##d/dx(sin(x)(sin(x)+cos(x)))=cos(2x)+sin(2x)##