Let \(n\) be a positive integer. The polynomial \(\mathrm{p}\) is defined by the identity
\[\mathrm{p}(\cos\theta) \equiv \cos\big((2n+1)\theta\big) + 1\,.\]
Show that
\[\cos\big((2n+1)\theta\big) = \sum_{r=0}^{n} \binom{2n+1}{2r} \cos^{2n+1-2r}\theta\,(\cos^2\theta - 1)^r\,.\]
By considering the expansion of \((1+t)^{2n+1}\) for suitable values of \(t\), show that the coefficient of \(x^{2n+1}\) in the polynomial \(\mathrm{p}(x)\) is \(2^{2n}\).
Show that the coefficient of \(x^{2n-1}\) in the polynomial \(\mathrm{p}(x)\) is \(-(2n+1)2^{2n-2}\).
It is given that there exists a polynomial \(\mathrm{q}\) such that
\[\mathrm{p}(x) = (x+1)\,[\mathrm{q}(x)]^2\]
and the coefficient of \(x^n\) in \(\mathrm{q}(x)\) is greater than \(0\).
Write down the coefficient of \(x^n\) in the polynomial \(\mathrm{q}(x)\) and, for \(n \geqslant 2\), show that the coefficient of \(x^{n-2}\) in the polynomial \(\mathrm{q}(x)\) is
\[2^{n-2}(1-n)\,.\]