Number of values of $\theta$ in $\left[0,2\pi \right]$ for which at least two roots of equation 

$x^3-\left(1+\cos \theta +\sin \theta \right)x^2+\left(\cos \theta \sin \theta +\cos \theta +\sin \theta \right)x-\sin \theta cos\theta =0$ are equal is

Roots are 1, $\cos \theta and \sin \theta$ .Now two roots are equal 

$$\begin{gathered} \Rightarrow \sin \theta =1 or \cos \theta =1 or \sin \theta =\cos \theta \\ \Rightarrow \theta =\frac{\mathrm{\pi }}{2},\theta =0,2\pi or \theta =\frac{\mathrm{\pi }}{4}or \frac{5\mathrm{\pi }}{4} \end{gathered}$$

$\Rightarrow there are 5 values of \theta in\left[0,2\mathrm{\pi }\right].$