How did the Lone Star Republic get their independence from Mexico?

i know that texas became the lone star republic after getting their independence from mexico, but how?How did the Lone Star Republic get their independence from Mexico?Santa Anna was captured at San Jacinto. He traded his life to Houston for independence of Texas.How did the Lone Star Republic get their independence from Mexico?In 1820, Moises Austin requested mexican government its permission to settle in Texas.



In 1834, just 14 years later they initiated their "claim" to gain independence so for sure they didnt have rights to declare independent in a land owned by Mexico for hundreds of years.



Mexican president Santa Anna attacked them and won El Alamo battle. But when he was about to eliminate them, he was captured and obligued to recognize Texas independence, return slaves to Texas, and send his troops back to the south of el Bravo.