# Proof of a Chebyshev Identity

Reference: Wiki on Chebyshev Polynomials, Proof problem 6.

$$\huge T_n(x) = \frac{ ( x - \sqrt{ x^2 - 1} )^n + ( x + \sqrt{ x^2 - 1 } ) ^ n } { 2} .$$

Solution: $\text{From the above identity, we get}$ $\large{ T }_{ n }(\cos { \theta )= } \frac { { \left( \cos { \theta -\sqrt { \cos ^{ 2 }{\theta}- 1 } } \right) }^{ n }+{ \left( \cos { \theta +\sqrt { \cos ^{ 2 }{ \theta } -1} } \right) }^{ n } }{ 2 } =\frac { { \left( \cos { \theta -i\sin { \theta } } \right) }^{ n }+{ \left( \cos { \theta +i\sin { \theta } } \right) }^{ n } }{ 2 }$.

$\text{Using}$ De Moivre's Theorem, $\text{we get}$ $\large\frac { { \left( \cos { \theta -i\sin { \theta } } \right) }^{ n }+{ \left( \cos { \theta +i\sin { \theta } } \right) }^{ n } }{ 2 } =\frac { \cos { n\theta -i\sin { \theta +\cos { n\theta +i\sin { n\theta } } } } }{ 2 } =\cos { n\theta }$.

$\text{This turns out to be the definition of the}$ Chebyshev Polynomial of the first kind.

$\therefore\text{ Proved}$

Note by Swapnil Das
5 years, 2 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

Nice work. I have a doubt $\sqrt{1-cos^2(\theta)}=\sin(\theta)$. Can you please explain how you got $i\sin(\theta)$ and how $\sqrt{1+cos^2(\theta)}=i\sin(\theta)$.

- 5 years, 2 months ago

Yup, I don't understand how Swapnil wrote $\sqrt{1-\cos^2 \theta}$ and $\sqrt{1+\cos^2 \theta}$

- 5 years, 2 months ago

Bro, I am Swapnil. Is it fine now? (It was a typo)

- 5 years, 2 months ago

Yep, it is correct now. Sorry, Swaplin was a typo, I meant to type Swapnil :P :P :P

- 5 years, 2 months ago

Correct. That's a typo, thanks for pointing out!

- 5 years, 2 months ago