<s>[INST] Find the value of $\tan A + \tan B$, given values of $\frac{\sin (A)}{\sin (B)}$ and $\frac{\cos (A)}{\cos (B)}$ Given
$$\frac{\sin (A)}{\sin (B)} = \frac{\sqrt{3}}{2}$$
$$\frac{\cos (A)}{\cos (B)} = \frac{\sqrt{5}}{3}$$
Find $\tan A + \tan B$.
Approach
Dividing the equations, we get the relation between $\tan A$ and $\tan B$ but that doesn't help in getting the value of $\tan A + \tan B$. The value comes in terms of $\tan A$ or $\tan B$ but the expected answer is independent of any variable .
Also
$$\frac{\sin(A)\cdot\cos(B) + \sin(B)\cdot\cos(A)}{\cos(A)\cdot\cos(B)} = \tan(A) + \tan(B)$$
We could get a value only if instead of $\cos A$ there was $\sin B$ in the relation(which we get on adding the ratios)
 [/INST] Hint: 
Let  $\dfrac{\sin A}{\sqrt3}=\dfrac{\sin B}2=p$ and $\dfrac{\cos A}{\sqrt5}=\dfrac{\cos B}3=q$
$\implies(\sqrt3p)^2+(\sqrt5q)^2=1$
and  $(2p)^2+(3q)^2=1$
Solve for $p^2,q^2$
</s>