I'd like to solve analytically the following equation, where $\alpha_i$ and $\beta_i$ have known values in $\mathbb{R}$: \begin{equation} \sum_{i\leqslant N} \alpha_i\,\cos(\beta_i\,t)=0 \end{equation}
Would there be an exact solution to this problem?
Mathematica only finds approximated solutions, which prevents me from calculating the null space of the matrix whose determinant is the left-hand side of the above equation.