TITLE: Proof that $\cos(\pi/4)=\frac{\sqrt2}{2}$
QUESTION [6 upvotes]: Normally I just look up or remember that $\cos(\pi/4)=\frac{\sqrt2}{2}$, or type "$\cos(\pi/4)$" into WolframAlpha to check the answer.
But what about the first time someone wanted to know what $\cos(\pi/4)$ was? How could they find out before trig tables and WolframAlpha?
How can you prove that $\cos(\pi/4)=\frac{\sqrt2}{2}$?

REPLY [3 votes]: First of all 
$$\cos(x)=\sin(\frac{\pi}{2}-x)$$
therefore 
$$\cos(\frac{\pi}{4})=\sin(\frac{\pi}{2}-\frac{\pi}{4})=\sin(\frac{\pi}{4})$$
Secondly 
$$\sin^2(x)+\cos^2(x)=1$$
and thus
$$\sin^2(\frac{\pi}{4})+\cos^2(\frac{\pi}{4})=1\Leftrightarrow 2\cos^2(\frac{\pi}{4})=1\Leftrightarrow \cos(\frac{\pi}{4})=|\frac{\sqrt{2}}{2}|$$
Since $\frac{\pi}{4}$ belongs to the first quadrant then 
$$\cos(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$$