2. (i) Show, with working, that $$x ^ { 3 } - ( 1 + \cos \theta + \sin \theta ) x ^ { 2 } + ( \cos \theta \sin \theta + \cos \theta + \sin \theta ) x - \sin \theta \cos \theta$$ equals $$( x - 1 ) \left( x ^ { 2 } - ( \cos \theta + \sin \theta ) x + \cos \theta \sin \theta \right)$$ Deduce that the cubic in (1) has roots $$1 , \quad \cos \theta , \quad \sin \theta$$ (ii) Give the roots when $\theta = \frac { \pi } { 3 }$. (iii) Find all values of $\theta$ in the range $0 \leqslant \theta < 2 \pi$ such that two of the three roots are equal. (iv) What is the greatest possible difference between two of the roots, and for what values of $\theta$ in the range $0 \leqslant \theta < 2 \pi$ does this greatest difference occur? Show that for each such $\theta$ the cubic (1) is the same.
2. (i) Show, with working, that
$$x ^ { 3 } - ( 1 + \cos \theta + \sin \theta ) x ^ { 2 } + ( \cos \theta \sin \theta + \cos \theta + \sin \theta ) x - \sin \theta \cos \theta$$
equals
$$( x - 1 ) \left( x ^ { 2 } - ( \cos \theta + \sin \theta ) x + \cos \theta \sin \theta \right)$$
Deduce that the cubic in (1) has roots
$$1 , \quad \cos \theta , \quad \sin \theta$$
(ii) Give the roots when $\theta = \frac { \pi } { 3 }$.\\
(iii) Find all values of $\theta$ in the range $0 \leqslant \theta < 2 \pi$ such that two of the three roots are equal.\\
(iv) What is the greatest possible difference between two of the roots, and for what values of $\theta$ in the range $0 \leqslant \theta < 2 \pi$ does this greatest difference occur?
Show that for each such $\theta$ the cubic (1) is the same.\\