only need parts (c) and (d) 1. In this question, you will be using the following trigonometric identities:cos? a + sin a(1)(2)1cos(a + B)sin(a + B)cos a cos B – sin a sin 3sin a cos 3 + cos a sin Bwhere a, B E R. You do not need to prove these identities. You may also use withoutproof the fact that the set{ a eR}CO Asin ais exactly the set of unit vectors in R?.Now for any real number ,defineCO A– sin aRasin aCOS a(a) Prove that for all a, B E R,R.R3 = Ra+8(b) Using part (a), or otherwise, prove that Ra is invertible and that R = R-a, forall a E R.(c) Prove that for all a E R and all x, y E R²,(Rax) · (Ray) = x • y(d) Suppose A is a 2 x 2 matrix such that for all x, y e R²,(Ax) · (Ay) = x•yMust it be true that ARa, for some a E R? Either prove this, or give a||counterexample (including justification).

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In this question, you will be using the following trigonometric identities: cos? a + sin? a (1) (2) (3) 1 cos(a + B) sin(a + B) cos a cos 3 –- sin a sin 3 sin a cos B + cos a sin B where a, B E R. You do not need to prove these identities. You may also use without proof the fact that the set { CO a : αER sin a is exactly the set of unit vectors in R?. Now for any real number a, define CO a – sin a Ra sin a COS a (a) Prove that for all a, ß E R, R.R3 : Ra+ß (b) Using part (a), or otherwise, prove that Ra is invertible and that R1 all a E R. R-a, for (c) Prove that for all a ER and all x,y € R², (Rax) · (Ray) = x• y (d) Suppose A is a 2 x 2 matrix such that for all x, y e R?, (Ах) (Ау) — х:у Must it be true that A = Ra, for some a e R? Either prove this, or give a counterexample (including justification).