Let  $f\left(x\right)=\cos x(\sin x+\sqrt{{\sin }^{2}x+{\sin }^2\theta })$ where $\theta$ is a given constant, then maximum value of $f\left(x\right)$ is

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

$\Rightarrow {(f\left(x\right)\sec x-\sin x)}^2={\sin }^{2}x+{\sin }^2\theta$

$\Rightarrow f^2\left(x\right){\sec }^{2}x-2f\left(x\right)\tan x={\sin }^2\theta$

$\Rightarrow f^2\left(x\right){\tan }^{2}x-2f\left(x\right)\tan x+$$f^2\left(x\right)-{\sin }^2\theta =0$

$\therefore 4f^2\left(x\right)\ge 4f^2\left(x\right)(f^2\left(x\right)-{\sin }^2\theta )$

$\Rightarrow f^2\left(x\right)\le 1+{\sin }^2\theta \Rightarrow |f\left(x\right)\le \sqrt{1+{\sin }^2\theta }|$