Trigonometric equations

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Question

Number of values of $θ$ in $[0, 2 π]$ for which at least two roots of equation
$x^{3} - (1 + \cos θ + \sin θ) x^{2} + (\cos θ \sin θ + \cos θ + \sin θ) x - \sin θ c o s θ = 0$ are equal is

Moderate

Solution

Roots are 1, $\cos θ a n d \sin θ$ .Now two roots are equal
$\Rightarrow \sin θ = 1 o r \cos θ = 1 o r \sin θ = \cos θ \Rightarrow θ = \frac{π}{2}, θ = 0, 2 π o r θ = \frac{π}{4} o r \frac{5 π}{4}$
$\Rightarrow t h e r e a r e 5 v a l u e s o f θ i n [0, 2 π] .$

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