Question: 4. Using the substitution = 2 arctant, it is possible to transform any rational function of trig functions into a rational function of t,

4. Using the substitution = 2 arctant, it is possible to transform any rational function of trig functions into a rational function of t, and hence use partial fractions. In this problem, you will use this substitution to (almost) find an antiderivative of 1 cos + sin (a) Using the double-angle identity cos(24) = 2(cos ) 2 1, and a triangle representing 2 t 1 1 t2 tan = show that cos 0 = 2 1 = t2 1+ t (b) By using a double-angle identity for sin, find a similar formula for sin 0. (c) Find a formula for do using the chain rule. (d) Using your results above, find a new form of the integral [ 1 cos + sin de which does not reference any trigonometric functions. Simplify the integrand fully but do not integrate.

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a To find cos2 we can use the doubleangle identity for cosine cos2 2cos 1 Rearranging this equation ... View full answer

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