If $\sin \theta+\cos \theta=\sqrt{5} \sin (90-\theta)$, find the value of $\cot \theta$.

$\frac{\sqrt{5}-1}{5}$ $\frac{\sqrt{5}+1}{4}$ $\frac{\sqrt{5}+1}{3}$ $\frac{\sqrt{5}-1}{4}$

$\frac{\sqrt{5}+1}{4}$