If $\cos x-\sin x=\sqrt{a} \sin x$, then $a \sin x+\cos x-\sin x=$

$$ \text { Match the items of List-I to the items of List-II } $$

$$ \text { The correct match is } $$

If $\cot \left(\frac{A}{2}\right)=\sqrt{\frac{1+a}{1-a}} \cdot \cot \left(\frac{\theta}{2}\right)$, then $\cos \theta=$

If $\sin \theta \cosh \alpha=\tan x, \cos \theta \sinh \alpha=\sec x$, then $\cos 2 \theta \cosh 2 \alpha=$