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.

• The question is not quite clear. Are you trying to find values for $A$ and $B$ for which the two equations hold? – stewbasic Aug 19 '16 at 2:11
• What is your question? Can you review what you posted? – DeepSea Aug 19 '16 at 2:11
• @deepsea I did some edit to the question. – danny Aug 19 '16 at 2:45
• He is only asking for a hint, not for a full solution. No need to close... – imranfat Aug 19 '16 at 2:52
• Hint: square up and add those two: $sin A = \sqrt{3} / 2 \;sin B,\; cos A = \sqrt{5} / 2 \;cos B$. – dxiv Aug 19 '16 at 2:55

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?