How do I do this problem?

Question

Find number of values of x satisfying the equation cos ((4pi/3) - arccos(x))=x

Answer 1

Let α=4pi/3, let β=arccos x.  Then sin β = ±√(1-x²) unless you confine yourself to the first quadrant for β.

 

Then you view the problem as cos (α-β) = cosα cosβ + sinα sinβ.

 

Proceed, using the specific values of all quantities, and you will have an equation involving x and various numbers.

 

Note: there are infinitely many values for β, so be careful.  On the other hand, x is governed by |x| ≤ 1.