Chapter 7.4, Problem 59E

The German mathematician Karl Weierstrass (1815–1897) noticed that the substitution t = tan(x/2) will convert any rational function of sin x and cos x into an ordinary rational function of t.

(a) If t = tan(x/2), –π < x < π, sketch a right triangle or use trigonometric identities to show that

$\cos \left(\frac{x}{2}\right)=\frac{1}{\sqrt{1+t^2}}\text{and}\sin \left(\frac{x}{2}\right)=\frac{t}{\sqrt{1+t^2}}$

(b) Show that

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

(c) Show that

$dx=\frac{2}{1+t^2}dt$