\frac 1{\sin\angle TAC}+\frac 1{\sin\angle TBC}=\frac{TA}{TB_0}+\frac{TB}{TA_0}=2\left(\frac{TA_1}{TB_0}+\frac{TB_1}{TA_0}\right)\geq 2\left(\frac{TA_0}{TB_0}+\frac{TB_0}{TA_0}\right)\geq2\cdot 2\sqrt{\frac{TA_0}{TB_0}\cdot\frac{TB_0}{TA_0}}=4