= Solution
{c}
\b[The truncated <Fourier cosine series> gives the better approximation on the closed interval], in the senses of <uniform convergence> and the asymptotic <L2 norm> error. To make the comparison precise, let $C_N$ and $S_N$ retain the frequencies $1\leq n\leq N$, with the constant term also included in $C_N$.
The cosine <Fourier coefficients> are $O(n^{-2})$. In fact, <absolute convergence> gives the uniform tail bound
$$
\sup_{0\leq x\leq\pi}|f(x)-C_N(x)|
\leq\frac4\pi\sum_{n>N}\frac1{n^2}
\leq\frac4{\pi N}\longrightarrow0.
$$
For the <Fourier sine series>, every <Fourier partial sum> satisfies $S_N(0)=S_N(\pi)=0$. Thus
$$
\boxed{\sup_{0\leq x\leq\pi}|f(x)-S_N(x)|\geq1+\pi},
$$
so there is no <uniform convergence> to $f$ on the closed interval. Its <odd extension> has <jump discontinuities>, producing the <Gibbs phenomenon> near the endpoints; its <Fourier coefficients> decrease only like $1/n$.
The advantage persists in the <L2 norm>, which ignores individual endpoint values. By <Parseval's identity> and <Fourier orthogonality>,
$$
\|f-C_N\|_2^2=\frac\pi2\sum_{n>N}a_n^2=O(N^{-3}),\qquad
\|f-S_N\|_2^2=\frac\pi2\sum_{n>N}b_n^2=O(N^{-1}).
$$
The nonzero odd cosine coefficients and both parity classes of sine coefficients also give matching lower bounds of these orders by comparison with the corresponding <p-series>. Hence their <L2 norm> errors decrease respectively like $N^{-3/2}$ and $N^{-1/2}$. This is a comparison of approximation quality in specified norms, not a claim that one finite sum has smaller error at every individual interior point.
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