The range of $\cos \theta \left(\sin \theta +\sqrt{{\sin }^2\theta +{\sin }^2\alpha }\right)$ is

$y=\cos \theta \left(\sin \theta +\sqrt{{\sin }^2\theta +\sin \alpha }\right)$

$\frac{y}{\cos \theta }=\sin \theta +\sqrt{{\sin }^2\theta +{\sin }^2\alpha }$

$y\sec \theta -\sin \theta =\sqrt{{\sin }^2\theta +{\sin }^2\alpha }4y^2\left[1-y^2+{\sin }^2\alpha \right]\ge 0$

Squaring on both sides $\Rightarrow 1-y^2+{\sin }^2\alpha \ge 0$

$y^2{\sec }^2\theta +{\sin }^2\theta -2y\tan \theta ={\sin }^2\theta +{\sin }^2\alpha y^2-1-{\sin }^2\alpha \le 0$

$y^2+y^2{\tan }^2\theta -2y\tan \theta -{\sin }^2\alpha =0 y^2-\left(1+{\sin }^2\alpha \right)\le 0$

$\tan \theta \in R\Rightarrow \Delta \ge 0$

$4y^2-4.y^2.\left(y^2-{\sin }^2\alpha \right)\ge 0 y\in \left[-\sqrt{1+{\sin }^2\alpha },\sqrt{1+{\sin }^2\alpha }\right]$