For light to be trapped in a fiber, the angle θ0 at which it enters must obey

n0 sin θ0 < n1 sin (90o-θc)

Trig identities then tell us that
n0 sin θ0 < n1 cos θc
n0 sin θ0 < n1 [1 - sin2θc]1/2

But the sine of the critical angle is the ratio n2/n1, so

n0 sin θ0 < n1 [1 - (n2/n1)2]1/2
n0 sin θ0 <  [n12 - n22]1/2