Can we show this polynomial has roots $1$, $\cos\theta$ and $\sin\theta$?
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Ref: R8923

Question

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.