plz solve diffentiation problem

diffn by two methods
dy/dt

method #1
t=sin y
y=sin^-1 t

diff both side w.r.t 't'

dy/dt = 1/sqrt [1-t^2]

method #2
t=sin y
sin y=t
cos y.dy/dt=1
dy/dt=1/cos y

which one of two method is right and why??

t = sin y

sin² y + cos² y = 1

cos² y = 1 - sin² y

cos² y = 1 - t²

So both methods correct

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