Chapter 8.5, Problem 87E

Rational functions of trigonometric functions An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution u = tan (x/2) or equivalently x = 2 tan⁻¹ u. The following relations are used in making this change of variables.

$A:dx=\frac{2}{1+u^2}duB:\sin x=\frac{2u}{1+u^2}C:\cos x=\frac{1-u^2}{1+u^2}$

88. Verify relation A by differentiating x = 2 tan⁻¹ u. Verify relations B and C using a right-triangle diagram and the double-angle formulas

$\sin x=2\sin \frac{x}{2}\cos \frac{x}{2}\text{and}\cos x=2{\cos }^2\frac{x}{2}-1$