<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:
$$\tan A=\frac{\sin A}{\cos A}=\frac{\frac{\sqrt{3}}2\sin B}{\frac{\sqrt{5}}3\cos B}=\frac{3\sqrt{3}}{2\sqrt{5}}\tan B\tag{1}$$
And
$$1=\sin^2 A+\cos^2 A=\frac{3}{4}\sin^2 B+\frac{5}{9}\cos^2 B=\frac59\left(\sin^2 B+\cos^2B\right)+\frac7{36}\sin^2 B$$
Last equation implies
$$\sin^2B=\frac{16}7$$
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