Problem: Find a uniformly valid approximation to the solution of $\epsilon y'' + \cos(x)y' + y = -1$ with boundary conditions $y(0) = 1$, $y(1) = 1$ in the limit $\epsilon \rightarrow 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.
Solution: $$\boxed{y(x) = -1 + \frac{2(\sec(1) + \tan(1))}{\sec(x) + \tan(x)} + 2(1 - \sec(1) - \tan(1))e^{-x/\epsilon}}$$