The number of values of the parameter α ∈ [0, 2π] for which the...

For the quadratic function to be square of a linear function

$D = 0$
$⇒ 4\cos^{2} \alpha - 4 \cdot \dfrac{1}{2}sin \alpha\left(cos \alpha + sin \alpha\right) = 0 ⇒ \left(2cos \alpha + sin \alpha\right)\left(cos \alpha - sin \alpha\right) = 0$
$⇒ tan \alpha = 1$ or $- 2$
For $\alpha \in \left[0, 2\pi\right] there are two solutions for each.<br />$⇒$total four values of \alpha.$