α and β are the positive acute angles and satisfying equations 5sin⁡2β=3sin⁡2α and tan⁡β=3tan⁡α simultaneously. Then the value of tan⁡α+tan⁡β is ________ .

5×2tan⁡β1+tan2⁡β=3×2tan⁡α1+tan2⁡αor 5tan⁡β1+tan2⁡β=3tan⁡α1+tan2⁡α(i)Substituting tan⁡β=3tan⁡α we have 5×3tan⁡α1+9tan2⁡α=3tan⁡α1+tan2⁡α or 5+5tan2⁡α=1+9tan2⁡α or 4tan2⁡α=4 or tan⁡α=1, i.e., tan⁡β=3∴ tan⁡α+tan⁡β=4