The Dirichlet and Fejer Kernel Identities

In the setting of The Real Numbers: Standing Notation and Background, let $\cos$ and $\sin$ be the cosine and sine functions from $\mathbb{R}$ to $\mathbb{R}$, let $\pi$ be the real number of The Number Pi (i), let $2=1+1$, and for a real number $t$ let $t^{2}=tt$ as in The Real Numbers: Standing Notation and Background (i). Natural numbers are read in $\mathbb{R}$ through the canonical map fixed there, so that $N-m$ and $2N+1$ below denote real numbers. Finite sums are those of The Real Numbers: Standing Notation and Background (ii). Then the following hold for every $N\in\mathbb{N}$ and every $t\in\mathbb{R}$.

1. (The Dirichlet identity)(i)

$$\Bigl(1+2\sum_{m=1}^{N}\cos(2\pi mt)\Bigr)\sin(\pi t)=\sin\bigl((2N+1)\pi t\bigr).$$

2. (The Fejer identity)(ii)

$$\Bigl(N+2\sum_{m=1}^{N}(N-m)\cos(2\pi mt)\Bigr)\bigl(\sin(\pi t)\bigr)^{2}=\bigl(\sin(N\pi t)\bigr)^{2}.$$

The coefficient $N-m$ of the summand at $m=N$ is zero, so that summand vanishes and the sum involves only the cosines with $m<N$; writing the sum up to $N$ rather than up to $N-1$ avoids an empty sum when $N=1$.