In the paper, by means of the Faa di Bruno formula, with the help of explicit formulas for special values of partial Bell polynomials with respect to two specific sequences generated by derivatives at the origin of the inverse sine or inverse cosine, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes Maclaurin's series expansions for real powers of the inverse cosine (sine) functions and the inverse hyperbolic cosine (sine) functions. By comparing different series expansions for the square of the inverse cosine function and for the positive integer power of the inverse sine function, the author not only finds infinite series representations of the circular constant Pi and its powers, but also derives several combinatorial identities involving central binomial coefficients and the Stirling numbers of the first kind.