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1. In this question, you will be using the following trigonometric identities: cosa + sina = 1 (1) cos(a + 8) = cos a cos 8 - sin a sin B (2) sin(a + 8) = sin a cos 8 + cos a sin 8 (3) where a, B ER. You do not need to prove these identities. You may also use without proof the fact that the set :OER cos al {[: a {R} sin a is exactly the set of unit vectors in RP. Now for any real number a, define Ra = cosa - sina] sina COS O = % (a) Prove that for all a, 8 ER, R,R, = Rx+8 (b) Using part (a), or otherwise, prove that R, is invertible and that RiP =R_q, for all a ER (c) Prove that for all a € R and all x, y € R2, (Rox). (Ray) = xy (d) Suppose A is a 2 x 2 matrix such that for all x, y ERP, (Ax). (Ay) = xy = Must it be true that A Ra, for some a € R? Either prove this, or give a counterexample (including justification).