Correct Answer: $1:\sqrt{3}$

Solution : Given: $\alpha +\beta =90^\circ$ and $\alpha :\beta =2:1,$
Hence, $\frac{\alpha}{\beta} = \frac{2}{1}$
$\alpha = 2\beta$
Now, putting the values of $\alpha$, we get
$\alpha +\beta =90^\circ$
⇒ $2\beta+\beta =90^\circ$
⇒ $3\beta =90^\circ$
⇒ $\beta =30^\circ$
Putting the values of $\beta$, we get
$\alpha +\beta =90^\circ$
⇒ $\alpha + 30^\circ =90^\circ$
⇒ $\alpha =60^\circ$
Thus, the ratio of $\cos \alpha$ to $\cos \beta$
= $\cos 60^\circ$: $\cos 30^\circ$
= $\frac{1}{2}:\frac{\sqrt{3}}{2}$
= $1:\sqrt{3}$
Hence, the correct answer is $1:\sqrt3$.