List of Trigonometric Identities - Product-to-sum and Sum-to-product Identities

Product-to-sum and Sum-to-product Identities

The product-to-sum identities or prosthaphaeresis formulas can be proven by expanding their right-hand sides using the angle addition theorems. See beat (acoustics) and phase detector for applications of the sum-to-product formulæ.

Product-to-sum





$\begin{align} \prod_{k=1}^n \cos \theta_k & = \frac{1}{2^n}\sum_{e\in S} \cos(e_1\theta_1+\cdots+e_n\theta_n) \\ & \text{where }S=\{1,-1\}^n \end{align}$

Sum-to-product

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