Is it Correct to Divide Both Angles by 2 in Trigonometry?

[Born2Perform]

$$cos(2x)=2cos^2(x)-1$$
what algebrical rule guarantees me thet is right to divide both angles for 2:
$$cos(x)=2cos^2(x/2)-1$$

 

[uart]

what algebrical rule ...

Substitution, nothing more.

$$cos(2x)=2cos^2(x)-1$$

Let u=2x, then

$$cos(u)=2cos^2(u/2)-1$$

Now you can replace u with whatever pronumeral you desire.