Problem 4: We define a sequence of polynomials (T(z)) To(x)= 1, T₁(x) = x, and Tn(x) = 2xTn-1(x)-Tn-2(x), n>2. So T₂(x)=2x²-1, T3(x) = 42³-3x, etc. Show that T₁ (cos(t)) = cos(nt) for n20. [HINT: The trigonometric formula cos(x + y) + cos(x - y) 2 by cos(x) cos(y) = may come in handy, with a and y equal to suitable multiples of t. We will consider it known, so you do not have to prove this formula before using it.]
Problem 4: We define a sequence of polynomials (T(z)) To(x)= 1, T₁(x) = x, and Tn(x) = 2xTn-1(x)-Tn-2(x), n>2. So T₂(x)=2x²-1, T3(x) = 42³-3x, etc. Show that T₁ (cos(t)) = cos(nt) for n20. [HINT: The trigonometric formula cos(x + y) + cos(x - y) 2 by cos(x) cos(y) = may come in handy, with a and y equal to suitable multiples of t. We will consider it known, so you do not have to prove this formula before using it.]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Problem 4: We define a sequence of polynomials (T(x)) by
To(x)= 1, T₁(x) = x,
n=0
and
Tn(x) = 2xTn-1(x) - Tn-2(x), n > 2.
So T₂(x) = 2x2-1, T3(x) = 4x³-3x, etc. Show that
T, (cos(t)) = cos(nt)
for n20. [HINT: The trigonometric formula
cos(x) cos(y) =
cos(x+y)+cos(x - y)
2
may come in handy, with z and y equal to suitable multiples of t. We
will consider it known, so you do not have to prove this formula before
using it.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F16f04be2-15fb-4f26-8839-46a60b2d804c%2F0f3649a9-d513-411c-9869-1cfafc8db7a1%2Fj1hk34q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 4: We define a sequence of polynomials (T(x)) by
To(x)= 1, T₁(x) = x,
n=0
and
Tn(x) = 2xTn-1(x) - Tn-2(x), n > 2.
So T₂(x) = 2x2-1, T3(x) = 4x³-3x, etc. Show that
T, (cos(t)) = cos(nt)
for n20. [HINT: The trigonometric formula
cos(x) cos(y) =
cos(x+y)+cos(x - y)
2
may come in handy, with z and y equal to suitable multiples of t. We
will consider it known, so you do not have to prove this formula before
using it.]
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