Can you please help me so work out this question? I found this on Textbook but there is only answer, so I can not clearly understand. 1. In this question, you will be using the following trigonometric identities:cos? a + sin? a = 1cos(@ + B)sin(a + B)(1)(2)(3)cos a cos B – sin a sin 3sin a cos 3 + cos a sin 3where a, B E R. You do not need to prove these identities. You may also use withoutproof the fact that the set{cos a: α Rsin ais exactly the set of unit vectors in R?.Now for any real number a, definecos a - sin acos aRasin a(a) Prove that for all a, 3 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 € R?,(Rax) · (Ray) = x • y(d) Suppose A is a 2 × 2 matrix such that for all x, y € R²,(Ах) (Ау) —D х уMust it be true that A = Ro, for some a e R? Either prove this, or give acounterexample (including justification).

Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

1. In this question, you will be using the following trigonometric identities: cos? a + sin? a = 1 (1) %3D cos(a + B) = cos a cos 3 – sin a sin 3 sin(a + B) (2) sin a cos 3 + cos a sin 3 (3) %3D where a, B E R. You do not need to prove these identities. You may also use without proof the fact that the set [cos a :a € R sin a is exactly the set of unit vectors in R². Now for any real number a, define cos a - sin a Ra sin a cos a (a) Prove that for all a, ß E R, R,R3 = Ra+8 (b) Using part (a), or otherwise, prove that Ra is invertible and that R,' = R_a, for all a € R. (c) Prove that for all a E R and all x, y e R², (R.x) · (Ray) = x · y

1. In this question, you will be using the following trigonometric identities: cos? a + sin? a = 1 (1) %3D cos(a + B) = cos a cos 3 – sin a sin 3 sin(a + B) (2) sin a cos 3 + cos a sin 3 (3) %3D where a, B E R. You do not need to prove these identities. You may also use without proof the fact that the set [cos a :a € R sin a is exactly the set of unit vectors in R². Now for any real number a, define cos a - sin a Ra sin a cos a (a) Prove that for all a, ß E R, R,R3 = Ra+8 (b) Using part (a), or otherwise, prove that Ra is invertible and that R,' = R_a, for all a € R. (c) Prove that for all a E R and all x, y e R², (R.x) · (Ray) = x · y

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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