Half-Range Fourier Cosine Series/Identity Function/0 to Pi/Proof 2

Proof
Let $\map f x: \openint 0 \lambda \to \R$ be the identity function on the open real interval $\openint 0 \lambda$:
 * $\forall x \in \openint 0 \lambda: \map f x = x$

From Half-Range Fourier Cosine Series for Identity Function, the half-range Fourier cosine series for $\map f x$ can be expressed as:

The result follows by setting $\lambda = \pi$.