Solution to the Helmholtz equation with Dirichlet BCs for a square

Question

Attempting to solve the 2D Helmholtz, $\Delta u + k^2 u =0$ in a unit square domain defined by $[0,1]\times[0,1]$ with the following dirichlet boundary conditions: \begin{align} u(x_1,0) &= \cos(k x_1), \quad &\textrm{for} \quad 0 \leq x_1 \leq 1,\\ u(0,x_2) &= \cos(k), \quad &\textrm{for} \quad 0 \leq x_2 \leq 1,\\ u(x_1,1) &= \cos(k x_1), \quad &\textrm{for} \quad 0 \leq x_1 \leq 1, \label{HS BC2}\\ u(1,x_2) &= 1, \quad &\textrm{for} \quad 0 \leq x_2 \leq 1.\label{HS BC3} \end{align} I beleive this BVP results in the solution $u($x$)=\cos(k x_1)$, where x$=(x_1,x_2)$. But unfortunately I lost my notes on the derivation of this solution. Would anyone be able to help me out please?