Can you prove to me why cos^2(X) + sin^2(X) = 1?

The answer to this lies in the geometry of a circle. At GCSE you were taught that a circle has a radius (which I will call R) which is the distance from the centre to a point on the circle. We put the centre of this circle on the x-y plane at (0,0). Drawing a straight line (which I will call L) from the centre to a point (a,b) on the circle in the upper right quadrant, we have a distance which is also the radius R. Using Pythagoras Theorem we can say that a²+b² = R² ..(1)We now look at the angle between the x-axis and the line L. We shall call this angle X for convenience. Using SOH CAH TOA trigonometry from GCSE we have that cos(X) = a/R - which we rearrange to get: a = Rcos(X)sin(X) = b/R - which we rearrange to get b = Rsin(X)If we substitute these two equations into (1) then we get: R²cos²(X) + R²sin²(X) = R²We then divide both sides of this equation to get cos²(X) + sin²(X) = 1!