A point P(x,y) is first rotated anticlockwise about the origin through an angle α to P1 and then again anticlockwise through an angle β to P2.
a. (i) Obtain the coordinates of P2 in terms of x,y,α and β. Show all necessary steps.
b. (ii) Prove that P2 coincides with the position obtained by a single anticlockwise rotation of P through the angle (α+β).
c. (iii) If instead the second rotation were clockwise through the same angle α, what would be the coordinates of the final image of P? Simplify your answer.