NL-NWO 1969 Problem 5

a) Prove that for $n = 2,3,4,...$ holds: $$\sin a + \sin 2a + ...+ \sin (n-1)a=\frac{\cos a \left(\frac{a}{2}\right) - \cos \left(n-\frac{1}{2}\right) a}{2 \sin \left(\frac{a}{2}\right)}$$ b) A point on the circumference of a wheel, which, remaining in a vertical plane, rolls along a horizontal path, describes, at one revolution of the wheel, a curve having a length equal to four times the diameter of the wheel. Prove this by first considering tilting a regular $n$ -gon.