Prove $-\int_0^1 u_{xxx}u_x \eta = \int_0^1 (u_{xx})^2\eta - \int_0^1 \frac{1}{2} (u_x)^2 \eta_{xx}$ using Fourier series

Question

Let $u, \eta$ be smooth functions and $\eta$ compactly supported in $(0,1)$. Integrating by parts, we can easily prove $$-\int_0^1 u_{xxx}u_x \eta = \int_0^1 (u_{xx})^2\eta - \int_0^1 \frac{1}{2} (u_x)^2 \eta_{xx}$$

Can we obtain the same results using only the Fourier series expansion
$$u(x) = \sum_{n=1}^\infty \sqrt 2 \cos (\pi n x) \, a_n, \qquad \eta(x) = \sum_{n=1}^\infty \sqrt 2 \cos (\pi n x) \, b_n $$ as a tool?