How to simplify tan(cos^-1 x)

cos(arccos(x)) = x

sec(t)^2 - tan(t)^2 = 1

Therefore:

1 / cos(t)^2 - tan(t)^2 = 1
1 / cos(arccos(x))^2 - tan(arccos(x))^2 = 1
1 / x^2 - tan(arccos(x))^2 = 1
(1 - x^2) / x^2 = tan(arccos(x))^2
tan(arccos(x)) = (1/x) * sqrt(1 - x^2)