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 (x/2) = 1/(√1 +1 t2) and sin(x/2) = t/(√1+t2)
(b) Show that
cox x = (1 -t2)/(1 + t2) and sin x = 2t/(1 + t2)
(c) Show that
dx = 2/(1 + t2) dt