Consider the $2$-dimensional Laplace equation in polar coordinates $$u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0$$
on the region $r<1$ with $u(1,\theta)=\cos\theta$.
Normally the way that I would approach a problem like this would be to use seperation of variables, specifically I would let $u(r,\theta)=R(r)\Theta(\theta)$ and go from there.
However, here I'm told that I am able to look for a solution of the form $$u(r,\theta)=R(r)\cos\theta$$
Why can this be done?
As you said, you can use separation of variables. This will lead you to the equation $\Theta''(\theta)=-\Theta(\theta)$. This has a solution of the form $A\cos(\theta) + B\sin(\theta)$. But the boundary condition tells you that $B=0$.