In this paper, we calculate the Wronskian of the trigonometric system \[ \cos{\lambda_{1}x},\sin{\lambda_{1}x},\cos{\lambda_{2}x},\sin{\lambda_{2}x},\dots,\cos{\lambda_{n}x},\sin{\lambda_{n}x} \] and prove that this system is linearly independent when $\lambda_{k}\ne 0$ and $\lambda_{k}^{2}\ne \lambda_{l}^{2}$ for $k\ne l$, where $\lambda_{1},\lambda_{2},\dots,\lambda_{n}\in\mathbb{C}$, $n\in\mathbb{N}$ are constants and $x$ is a complex variable. By using it, we evaluate the determinant below \[ \left| \begin{array}{ccccccc} 1&0&1&0&\cdots&1&0\\ 0&1&0&1&\cdots&0&1\\ \lambda_{1}&0&\lambda_{2}&0&\cdots&\lambda_{n}&0\\ 0&\lambda_{1}&0&\lambda_{2}&\cdots&0&\lambda_{n}\\ \lambda_{1}^{2}&0&\lambda_{2}^{2}&0&\cdots&\lambda_{n}^{2}&0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&\lambda_{1}^{n-2}&0&\lambda_{2}^{n-2}&\cdots&0&\lambda_{n}^{n-2}\\ \lambda_{1}^{n-1}&0&\lambda_{2}^{n-1}&0&\cdots&\lambda_{n}^{n-1}&0\\ 0&\lambda_{1}^{n-1}&0&\lambda_{2}^{n-1}&\cdots&0&\lambda_{n}^{n-1} \end{array} \right|. \]