The formula you've shared appears to be a trigonometric identity that combines products of sines and cosines with summation over binary variables, likely related to group theory, combinatorics, or higher-level trigonometric identities. Such expressions often arise in advanced mathematical contexts such as Fourier analysis or quantum mechanics. They also resemble some formulas used in special function theory or signal processing due to the combination of product terms and trigonometric functions like sine and cosine.

Name of Discoverer:

It's challenging to pinpoint a single "discoverer" for such complex identities, as they build on work from many mathematicians over centuries. However, identities like these are often derived from Euler's work on trigonometric functions, as well as expansions in harmonic analysis and Fourier transforms.

Resources and Videos:

1. Euler's Trigonometric Formulas – Euler's contributions to trigonometric identities and exponential functions, particularly the famous Euler's formula, are foundational for understanding these identities. Videos explaining this can be found on platforms like YouTube or educational sites like Nagwa.

2. Sum-to-Product and Product-to-Sum Formulas – These transformations express products of sines and cosines in terms of sums, or vice versa, which are fundamental tools when dealing with expressions like yours. These can be explored in resources on Mathematics LibreTexts.

3. Combinatorial Trigonometric Sums – For deeper insights into sums over binary variables (like $e \in \{1, -1\}^n$), you may want to investigate combinatorial or harmonic analysis texts. This concept appears in Fourier analysis and other mathematical areas like coding theory.

4. Videos on Fourier Series and Transform – Understanding the breakdown of periodic functions using sines and cosines helps in visualizing formulas involving such sums and products. Search for Fourier series and Fourier transforms on learning platforms like Khan Academy or MIT OpenCourseWare.

Would you like more detailed explanations or further examples?

Related Questions:

1. What is Euler's formula and how does it relate to trigonometry?
2. How can product-to-sum formulas simplify trigonometric expressions?
3. What are common applications of Fourier transforms in signal processing?
4. What role do combinatorial sums play in trigonometric identities?
5. How do sine and cosine products appear in quantum mechanics equations?

Tip:

Understanding the derivation of product-to-sum formulas can give you a strong foundation to tackle more advanced trigonometric identities like this one.