I put the images to make it easier to follow. I don't understand why $\cos \alpha_{n+1} = \cos \frac{\alpha_n}{2}$ implies that $\alpha_n = \frac{\alpha_0}{2^n}$. I also don't understand why he changes it to $\arccos$. I have no problems with the other things. Can someone please clarify and show this steps better. This is from the book "Problem Solving Strategies" by Arthur Engel, E6 in the Invariance Principle section. Pages 3-4
You set $x_n/y_n=\cos\alpha_n$ for some $\alpha_n\in(0,\pi/2)$, since you know that $x_n/y_n\in(0,1)$. If $\cos A=\cos B$ for $A,B\in(0,\pi/2)$, necessarily $A=B$ (two angles in the first quadrant with equal cos have to be equal).
Therefore $\alpha_{n+1}=\alpha_n/2$, and then, recursively, $$ \alpha_{n}=\alpha_{n-1}/2=\alpha_{n-2}/2^2=\alpha_{n-3}/2^3=\dots=\alpha_0/2^n $$ since $x_n/y_n=\cos\alpha_n$ and $\alpha_n\in(0,\pi/2)$, $$ \alpha_n=\arccos(x_n/y_n), \qquad \alpha=0,1,2,\dots $$