Find number of possible values satisfying $\cos \left(\pi\sqrt{x-4}\right)\cos \left(\pi\sqrt{x}\right)=1$

Question

Find number of possible values satisfying the equation: $$\cos \left(\pi\sqrt{x-4}\right)\cos \left(\pi\sqrt{x}\right)=1$$

We are required to find the number of possible solutions for $x\in\Bbb R$.

I can't understand how to approach this.

Answer 1

Hint:

Since $\cos \theta \in [-1,1]$, and we're given the equation

$$\cos(\pi \sqrt{x - 4}) \cos(\pi \sqrt x) = 1$$

then either

$$\cos(\pi \sqrt{x - 4}) = \cos(\pi \sqrt x) = 1$$

or

$$\cos(\pi \sqrt{x - 4}) = \cos(\pi \sqrt x) = -1$$

Don't forget to invoke the periodicity of cosine as well.