Use De Moivre's theorem to prove that for any positive integer \(k > 1\),
\[ \sin(k\theta) = \sin\theta\cos^{k-1}\theta \left( k - \binom{k}{3}(\sec^2\theta - 1) + \binom{k}{5}(\sec^2\theta - 1)^2 - \cdots \right) \]
and find a similar expression for \(\cos(k\theta)\).
Let \(\theta = \cos^{-1}(\frac{1}{a})\), where \(\theta\) is measured in degrees, and \(a\) is an odd integer greater than \(1\).
Suppose that there is a positive integer \(k\) such that \(\sin(k\theta) = 0\) and \(\sin(m\theta) \neq 0\) for all integers \(m\) with \(0 < m < k\).
Show that it would be necessary to have \(k\) even and \(\cos(\frac{1}{2}k\theta) = 0\).
Deduce that \(\theta\) is irrational.
Show that if \(\phi = \cot^{-1}(\frac{1}{b})\), where \(\phi\) is measured in degrees, and \(b\) is an even integer greater than \(1\), then \(\phi\) is irrational.