Let $M=\begin{bmatrix} 0&\sin(x)&\sin(2x)&\sin(3x)&\sin(4x) \\\sin(x)&0&\sin(x)&\sin(2x)&\sin(3x) \\\sin(2x)&\sin(x)&0&\sin(x)&\sin(2x) \\\sin(3x)&\sin(2x)&\sin(x)&0&\sin(x) \\\sin(4x)&\sin(3x)&\sin(2x)&\sin(x)&0 \end{bmatrix}$.
Question 1. (out of 3) of the exercise asks for an easy eigenvalue and eigenvector, but after having searched for an hour, tried some Chebichev tricks, I haven't found anything.
Looking at simple cases (n=1,n=2), I didn't see any logical progression either.
By the way : Question 2 is to give its characteristic polynomial, so we are not supposed to use it here I guess.
Your matrix is real, symmetric, persymmetric, centrosymmetric, Toeplitz, and traceless. Let $J$ be the flip matrix (zeros except for 1's along the counter-diagonal). It is straightforward to show that the eigenvectors are either even or odd (with respect to elements about the middle of the vector). Therefore, there will be 3 even eigenvectors, and 2 odd eigenvectors. The reduced $2\times2$ problem for the odd eigenvectors is $$ \begin{bmatrix} -\sin4x & \sin x-\sin3x \\ \sin x-\sin3x & -\sin2x \end{bmatrix} v = \lambda v$$ You can straightforwardly find the eigenvalues and eigenvectors of this $2\times 2$ matrix. You can do a similar thing with the even eigenvectors, but you have a $3x3$ matrix. For the second part of the problem, you can see that the 5th degree characteristic equation should factor into the product of a quadratic and cubic, based on the arguments above. None of this seems like an "easy" answer, so this is probably not the right approach.
Taking a more analytic approach, notice the matrix $M$ is the imaginary part of the discrete Fourier convolution operator of order 5 of a delta function. I'm not sure if this helps.