Using a Suitable substitution or otherwise, find the differential of y= arctan(sinxcosx), in terms of y and x.

First of all, replace sinxcosx with 1/2 sin2x. Then you should let U=1/2 Sin2x and replace that in the formula. If y=arctan(U), then U=tany. work out dU/dy which is Sec²y. Using the trigonometric identity sin²y + cos²y= 1, sec²y= 1+tan²y. The differential now becomes 1+U². Flip the equation around to give dy/dU = 1/(1+U²).to get the differential in terms of y and x first replace U² with 1/4 sin²2x. using chain rule, dy/dx=dy/du * du/dx. du/dx = cos2x, so combining the two equations dy/dx = cos2x/(1 + 1/4 sin²x) which can be simplified to dy/dx = 4cos2x/(4 + sin²2x)