What is the usual way of proving things like
$$\cos\left(\frac{\pi}{5}\right)=\frac{1}{2}\phi?$$
I know that there is an identity which claims the above, but how was it derived? Are other identities used in the process? I am interested in knowing this.
For example, I could follow through as such:
$$\cos\left(\frac{\pi}{5}\right)=\cos\left(-\frac{\pi}{5}\right)=-\cos\left(\pi-\frac{\pi}{5}\right)=-\cos\left(\frac{4\pi}{5}\right)=1-2\cos^2\left(\frac{2\pi}{5}\right),$$
which gets me somewhat close to the golden ratio, but I must admit that I am stuck here.
Do you guys have any ideas?
Best Answer
I don't know about the "usual" way to do this, but here's a geometric proof: draw an isosceles triangle $ABC$ with angles $\pi/5$, $\pi/5$, and $3\pi/5$, this last angle being at $A$. Now draw $AD$ such that $D$ is on $BC$ and angle $BAD$ measures $\pi/5$. Notice both small triangles $DAB$ and $DAC$ are isosceles. Let $AB = AC = DC = 1$ and $BC = x$. Then since triangle $DAB$ is similar to triangle $ABC$, $BD/AB = CA/BC$, or $(x-1)/1 = 1/x$. Hence $x = \phi$. Now $\cos \pi/5 = \cos\angle B = \text{adj} / \text{hyp} = (BC/2) / AB = \phi /2$.
EDIT: darn, the image quality is atrocious.