# Wave Walker DSP

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DSP for Beginners: Simple Explanations for Complex Numbers! The second edition includes a new chapter on complex sinusoids.

Trigonometric Identity: sin(x)^2 + cos(x)^2 = 1
March 9, 2022

#### Introduction

In this post I derive a common trigonometric identity sin(x)^2 + cos(x)^2 = 1 using a form of Euler’s formula.

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(1)

#### Trigonometric Identity Derivation

Sine can be written as complex exponentials using a form of Euler’s formula (reference)

(2)

therefore sine squared is

(3)

Cosine can be written as complex exponentials using a form of Euler’s formula (reference)

(4)

therefore cosine squared is

(5)

(6)

therefore

(7)

You can find a list of additional trigonometric identities here.

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### 2 Responses

1. Bklolo says:

Hi, really enjoying the site — I’ll be purchasing the book soon enough!

Just wanted to point out that for item (3) it is covering part of the equation and the part that it is covering is missing the 2 from the FOIL, I believe.

1. Thank you for the comment and great feedback, very much appreciated!

Are you viewing the website on mobile? Try rotating your mobile device to landscape mode. I was able to see (3) on my mobile device once I did that. Also, the -2 comes from the terms – exp(j theta) exp(-j theta) – exp(-j theta) exp(j theta) = -1 -1 = -2.

Feel free to drop me a line at matt@wavewalkerdsp.com after you get a copy of the book, I’d love to get your feedback: what you liked, what you didn’t, and what I can add to the next edition to make it better.

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