place vertices of square at $(1,1), (-1,1) (-1,-1), (1,-1)$, and O at $(x,y)$ Write down expressions for the tangent of each of the relevant angles, and use $tan(a+b)$ formula (3 times) to get the tangent of the sum of these angles = $T = \frac{4xy(y^2-x^2)}{(y^2-x^2)^2-4x^2y^2+4}$ The denominator of the expression for T is obviously positive for x,y magnitude less than 1, so we can cross-multiply without reversing the inequality sign, and finally use AM/GM $2xy \le x^2+y^2$ to show that T is in the range -1 to +1, as required. Not very beautiful, but it works Merlin