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

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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)

James P.

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James P. is an online Further Mathematics tutor with MyTutor studying at Bath University

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