Hint: Use Euler's formula to rewrite the integrand in terms of $e^{i \theta}$, then rewrite the integral as a contour integral in the complex plane (what kind of shape does $e^{i \theta}$ parameterize?).
As for the intuition...
The $\cos \theta$ brings back fond memories of polar coordinates (remember $x = r \cos \theta$, $y = r \sin \theta$?), and then the $0$ to $2 \pi$ integral sparks the thought that somewhere, somehow, a circle is being parameterized. The quantity $e^{i \theta}$ does exactly that--it parameterizes the unit circle.
I remember from the hymns of ages past that Euler's formula can be used to rewrite $\cos \theta$ in terms of $e^{i \theta}$, so I follow in the footprints of the ancients and take advantage of this to rewrite the integrand. Calling upon the dark arts, I make a change of variables which transports me from the realm of the (real) line to the realm of the infinite (complex) plane. (The substitution $z = e^{i \theta}$ looks nice.)
Now that I'm in my element, I can go at the problem using all the magicks which were, until now, forbidden (Cauchy's theorem, the residue theorem, etc.). Of course I have my trusty emergency pack ready, as I might need some extra tools along the way (partial fractions).