I cannot follow your calculations, but here's my solution:
The derivative of f(x) = tan(x) is
f'(x) = 1 - tan^2(x) = 1 - sin^2(x) / (1 - sin^2(x))
The derivative of g(x) = arcsin(x) is
g'(x) = 1 / sqrt(1 - x^2)
Since y = f(g(x)) the derivative of y is
y' = f'(g(x)) * g'(x) =
(1 - x^2/(1-x^2) * 1 / sqrt(1-x^2)
Simplify and you get the answer from the book.