- Show, with working, that
\[\begin{equation}
x^3 - (1 + \cos\theta + \sin\theta)x^2 + (\cos\theta\sin\theta + \cos\theta + \sin\theta)x - \sin\theta\cos\theta, \qquad \label{eq:1}
\end{equation}\]
equals \[(x - 1)(x^2 - (\cos\theta + \sin\theta)x + \cos\theta\sin\theta).\] Deduce that the cubic in \(\eqref{eq:1}\) has roots \[1, \qquad \cos\theta, \qquad \sin\theta.\]

Give the roots when \(\theta = \dfrac{\pi}{3}\).

Find all values of \(\theta\) in the range \(0 \leq \theta < 2\pi\) such that two of the three roots are equal.

What is the greatest possible difference between two of the roots, and for what values of \(\theta\) in the range \(0 \leq \theta < 2\pi\) does this greatest difference occur?

Show that for each such \(\theta\) the cubic \(\eqref{eq:1}\) is the same.