# Wave Walker DSP

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Product to Sum Trigonometric Identities with Euler’s Formula
June 15, 2022

#### Introduction

This blog derives the product to sum trigonometric identities using Euler’s formula,

(1)

(2)

(3)

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#### cos(a) cos(b)

Start by substituting Euler’s formula in for both cosine terms,

(4)

Multiplying the right hand terms of (4),

(5)

Simplifying (5) with Euler’s formula results in

(6)

(7)

such that

(8)

#### sin(a) sin(b)

Start by substituting Euler’s formula in for both sine terms,

(9)

Multiplying the right hand terms of (9),

(10)

Simplifying (10) with Euler’s formula results in

(11)

(12)

such that

(13)

#### cos(a) sin(b)

Start by substituting Euler’s formula in for cosine and sine,

(14)

Multiplying the right hand terms of (14),

(15)

Simplifying (15) with Euler’s formula results in

(16)

(17)

such that

(18)

#### Conclusion

The product to sum trigonometric identities were proven using Euler’s formula.

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