Number of values of the parameter $\mathrm{\alpha }\in [0,2\mathrm{\pi }]$  for which  the quadratic function, $(\sin \mathrm{\alpha }){\mathrm{x}}^2+2\cos \mathrm{\alpha x}+\frac{1}{2}(\cos \mathrm{\alpha }+\sin \mathrm{\alpha })$ is the square of a linear function is

Given expression is a perfect square if Disc = 0

$\begin{array}{l}\Rightarrow 4{\cos }^2\mathrm{\alpha }-2(\cos \mathrm{\alpha }+\sin \mathrm{\alpha })\sin \mathrm{\alpha }=0\\ \Rightarrow 4{\cos }^2\mathrm{\alpha }-2{\sin }^2\mathrm{\alpha }-2\sin \mathrm{\alpha cos}\mathrm{\alpha }=0\\ \Rightarrow {\tan }^2\mathrm{\alpha }+\tan \mathrm{\alpha }-2=0\\ \Rightarrow (\tan \mathrm{\alpha }-1)(\tan \mathrm{\alpha }+2)=0\Rightarrow \mathrm{\alpha }=\frac{\mathrm{\pi }}{4}\text{ or }\mathrm{\pi }-{\tan }^{-1}\left(2\right)\end{array}$