List of Trigonometric Identities - Lagrange's Trigonometric Identities

Lagrange's Trigonometric Identities

These identities, named after Joseph Louis Lagrange, are:

$\begin{align} \sum_{n=1}^N \sin n\theta & = \frac{1}{2}\cot\frac{\theta}{2}-\frac{\cos(N+\frac{1}{2})\theta}{2\sin\frac{1}{2}\theta}\\ \sum_{n=1}^N \cos n\theta & = -\frac{1}{2}+\frac{\sin(N+\frac{1}{2})\theta}{2\sin\frac{1}{2}\theta} \end{align}$

A related function is the following function of x, called the Dirichlet kernel.

$1+2\cos(x) + 2\cos(2x) + 2\cos(3x) + \cdots + 2\cos(nx) = \frac{\sin\left(\left(n +\frac{1}{2}\right)x\right)}{\sin(x/2)}.$

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