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2. (i) Given sin(x + y) = sin(x) cos(y) + cos(x) sin(y), prove the double angle formula sin(r) = 2 sin(x/2) cos(x/2) (ii) Given cos(r + y) = cos r cos y – sin r sin y and the result of Exercise 2 (i), show cos(3r) = cos(x) – 4 sin²(x) cos(x). (iii) Prove that if r is a rational number, then r – v2 is irrational. (Use a contradiction argument)

2. (i) Given sin(x + y) = sin(x) cos(y) + cos(x) sin(y), prove the double angle formula sin(r) = 2 sin(x/2) cos(x/2) (ii) Given cos(r + y) = cos r cos y – sin r sin y and the result of Exercise 2 (i), show cos(3r) = cos(x) – 4 sin²(x) cos(x). (iii) Prove that if r is a rational number, then r – v2 is irrational. (Use a contradiction argument)

Advanced Engineering Mathematics
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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Can you please help me with i, ii, and iii of the attached problem. Thank you. 

2. (i) Given sin(x + y) = sin(x) cos(y) + cos(x) sin(y), prove the double angle formula sin(x) = 2 sin(x/2) cos(x/2) (ii) Given cos(x + y) = cos x cos y – sin r sin y and the result of Exercise 2 (i), show cos(3x) = cos(x) – 4 sin? (x) cos(x). (iii) Prove that if r is a rational number, then r - V2 is irrational. (Use a contradiction argument) (iv) We say {fn} converges uniformly to f(x) iff (Ve > 0)(3n > 0)(Vx)(n >n= |fn (x) – f(x)| < e). What does it mean to say that {fn} does not converge uniformly to f? (Negate the statement)
Transcribed Image Text:2. (i) Given sin(x + y) = sin(x) cos(y) + cos(x) sin(y), prove the double angle formula sin(x) = 2 sin(x/2) cos(x/2) (ii) Given cos(x + y) = cos x cos y – sin r sin y and the result of Exercise 2 (i), show cos(3x) = cos(x) – 4 sin? (x) cos(x). (iii) Prove that if r is a rational number, then r - V2 is irrational. (Use a contradiction argument) (iv) We say {fn} converges uniformly to f(x) iff (Ve > 0)(3n > 0)(Vx)(n >n= |fn (x) – f(x)| < e). What does it mean to say that {fn} does not converge uniformly to f? (Negate the statement)
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