<s>[INST] If $\frac{\sin A}{\sin B}=\frac{\sqrt{3}}{2}$ and $\frac{\cos A}{\cos B}=\frac{\sqrt{5}}{2}$, then $\tan A+\tan B = $??? 
$$\text{If}\quad \frac{\sin A}{\sin B}=\frac{\sqrt{3}}{2} \quad\text{and}\quad \frac{\cos A}{\cos B}=\frac{\sqrt{5}}{2}\,, \quad\text{then}\quad \tan A+\tan B = \text{???}$$
  Here, $0<A,B<\frac\pi2$.

I tried many times but I am getting no result. Please help with some hint.
 [/INST] hint: $1 = \dfrac{3\sin^2B+5\cos^2B}{4}=\dfrac{5-2\sin^2B}{4}\implies \sin B = \dfrac{1}{\sqrt{2}}$. Can you continue to the finish line?
</s>