They are identically eqivilent - in fact when you wish to prove d[arcsin(x)]/dx = 1/sqrt(1-x²) you use your method #2 in order to do so. Then you do what Glaysher has done to express explicitly, and cleverly discard the negative sqrt.

Look at what you have done.

Method #1 differentiates y w.r.t. t and your answer is in terms of t. This is called expressing the function EXPLICITLY.

Method #2 differentiates y w.r.t. t and your answer is in terms of y. This is called expressing the function IMPLICITLY.

All the best, Matt