Evaluate $\sin(\frac{\pi}{8})$ and $\cos(\frac{\pi}{8})$ 
Evaluate $\sin(\frac{\pi}{8})$ and $\cos(\frac{\pi}{8})$

I was just wondering what I am doing wrong, as I don't seem to be arriving at the correct answer for $\sin(\frac{\pi}{8})$
What I did:
Let $\theta = \frac{\pi}{8}$
$\cos(2\theta) = 2\cos^2(\theta) - 1$
$\therefore \cos(\theta) = \sqrt{\frac{\cos(2\theta) + 1}{2}} = \frac{\sqrt{\sqrt{2} + 2}}{2}$
Now, $\sin(2\theta) = 2\cos(\theta)\sin(\theta)$
$\therefore \sin(\theta) = \frac{\sin(2\theta)}{2\cos(\theta)}$
Solving for $\sin(2\theta)$ and substituting in my answer for $\cos(\theta)$, I get:
$\frac{\sqrt{2}}{2\sqrt{\sqrt{2} + 2}}$ but I have an answer saying that $\sin(\frac{\pi}{8}) = \frac{\sqrt{2 - \sqrt{2}}}{2}$ and I couldn't seem to arrive at that.

The trick is to rationalize the denominator by multiplying by the conjugate, along the following lines:
$$\begin{align}\frac{\sqrt{2}}{2\sqrt{\sqrt{2} + 2}}&=\frac{\sqrt{2}}{2\sqrt{\sqrt{2} + 2}}\cdot\frac{\sqrt{2-\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\\&=\dfrac{\sqrt{2}\sqrt{2-\sqrt{2}}}{2 \sqrt {4-2}}=\dfrac{\sqrt{\not 2}\sqrt{2-\sqrt 2}}{2 \sqrt {\not 2} }\\\ &=\frac{\sqrt{2-\sqrt{2}}}{2}\end{align}$$ 
