A curve has parametric equations

$\displaystyle x=\theta + sin\theta , y = 1 + cos\theta, (0 \leq \theta \leq \pi)$.

Show that

$\displaystyle (\frac{dx}{d\theta})^2 + (\frac{dy}{d\theta})^2 = 4cos^2(\frac{\theta}{2})$

I differentiate the parametric equations, square them and get

$\displaystyle cos^2\theta +2cos\theta + 1 + sin^2\theta$, and I know that somehow I need to show that it's equal to $\displaystyle 4cos^2(\frac{\theta}{2}$ but I have no idea - I've looked at some basic trigonometric identities to simplify it but I get nowhere. Is there anyone who can help me with this?

Thanks if you have the time