= Solution
{c}
The <Fourier cosine series> on $[0,L]$ is
$$
f(x)\sim\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos\frac{n\pi x}{L},
\qquad
\boxed{a_n=\frac2L\int_0^Lf(x)\cos\frac{n\pi x}{L}\,dx\quad(n\geq0).}
$$
It is the <Fourier series> of the <even extension> to $[-L,L]$, repeated with <period> $2L$. The <Fourier sine series> is
$$
f(x)\sim\sum_{n=1}^\infty b_n\sin\frac{n\pi x}{L},
\qquad
\boxed{b_n=\frac2L\int_0^Lf(x)\sin\frac{n\pi x}{L}\,dx\quad(n\geq1).}
$$
It comes from the <odd extension> and its $2L$-<periodic extension>. These <Fourier coefficient> formulas follow by <Fourier orthogonality>, since every nonconstant <sine> or <cosine> has squared <integral> $L/2$ on $[0,L]$, while the <constant function> has squared <integral> $L$.
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