1. In this question, you will be using the following trigonometric identities: cosa + sin² a 1 (1) cos(a + 3) = cos a co

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1 In This Question You Will Be Using The Following Trigonometric Identities Cosa Sin A 1 1 Cos A 3 Cos A Co 1 (57.09 KiB) Viewed 44 times

1. In this question, you will be using the following trigonometric identities: cosa + sin² a 1 (1) cos(a + 3) = cos a cos 3-sin a sin 8 (2) sin(a+3)= sin a cos B+ cosa sin 3 (3) where a,BR. You do not need to prove these identities. You may also use without proof the fact that the set COS C à E sina is exactly the set of unit vectors in R². Now for any real number a, define cos a-sina Ra sin a cosa (a) Prove that for all a, 3 € R. Ra Ra Ra (b) Using part (a), or otherwise, prove that R. is invertible and that R¹ = R-a. for all a ER. (c) Prove that for all o R and all x, y € R², (Rox) (Ray)=x-y (d) Suppose A is a 2 x 2 matrix such that for all x, y € R², (Ax)-(Ay)=x-y Must it be true that A= R. for some a ER? Either prove this, or give a counterexample (including justification). (e) Let B= -[1] be any 2 x 2 matrix. cos a (i) Show that there are real numbers and a such that M11 sine (ii) Let a R. Use the invertibility of R. to prove that there are unique 12,22 ER such that cos a sin o D- =12 +₂2 sin a cos a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B=R₂U for some o R and some upper-triangular matrix U. (iv) Suppose that B= R₂U R&V, where a, ß ER and U and V are upper- triangular. Prove that if B is invertible, then U=+V. -