sin(sin^(-1)(x)) = x for all x in [-1, 1] — this always simplifies cleanly. But sin^(-1)(sin(x)) = x ONLY when x in [-pi/2, pi/2]. For x outside this interval, reduce using: sin^(-1)(sin(x)) = pi - x when x in [pi/2, 3pi/2], and = x - 2pi when x in [3pi/2, 5pi/2], etc. Similarly, cos^(-1)(cos(x)) = x for x in [0, pi], and = 2pi - x for x in [pi, 2pi]. tan^(-1)(tan(x)) = x for x in (-pi/2, pi/2), and = x - pi for x in (pi/2, 3pi/2). The graph of sin^(-1)(sin(x)) is a zigzag (triangle wave) with period 2pi and amplitude pi/2.