RecurrenceTable:
RecurrenceTable[{ x[n + 1] == x[n] + (1/2) (Sqrt[1 - y[n]^2] - x[n]), y[n + 1] == y[n] + (1/2) (Sqrt[1 - x[n + 1]^2] - y[n]), x[0] == 0, y[0] == 0}, {x, y}, {n, 1, 5}] // N
{{0.5, 0.433013}, {0.700694, 0.573237}, {0.760042, 0.611556}, {0.775621, 0.621377}, {0.779567, 0.623848}}
Credit goes to Stelios for spotting a mistake in the original answer :)