 Solution

Every finite sum of the sine modes is odd, and so is a limit in the L2 norm of such sums. In contrast, $x^2-1$ is a nonzero even function. Its inner product with every odd sine mode vanishes, so all its proposed Fourier sine series coefficients would be zero. It cannot be represented by this basis on the symmetric interval. An even Fourier cosine series is the appropriate alternative there.