arccos(x) = arcsec(x) Solve the equation for x

The function at the left side, arccos(x), is defined at [-1, 1], because it is the range of cos(x).  The function at the right side, arcsec(x), is defined at (-oo, -1] uu [1, +oo], because it is the range of sec(x).
Thus there are only two points at which both sides are defined, x_1 = -1 and x_2 = 1. Let's check whether they are the solutions: arccos(-1) = pi and arcsec(-1) = pi, arccos(1) = 0 and arcsec(1) = 0.
Yes, they are the solutions. The answer: x_1 = -1, x_2 = 1.