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)

Something is wrong. I calculated
\begin{align}
   \sin A &= \dfrac{\sqrt 3}{2}\sin B \\
   \cos A &= \dfrac{\sqrt 5}{3}\cos B \\
   \sin^2 A + \cos^2 A &= \dfrac 34 \sin^2 B + \dfrac59 \cos^2 B\\
   1 &= \dfrac 34 - \dfrac{7}{36} \cos^2 B\\
   \cos^2 B = -\dfrac{9}{7}
\end{align}
Which is impossible.
