To find:
The x-values at the intersection of the graphs of two functions.
Solution:
Two functions are:
[tex]y=\cos2x\text{ and }y=3\cos^2x-\sin^2x[/tex]
The functions are equal at the intersection. So,
[tex]\cos2x=3\cos^2x-\sin^2x[/tex]
The solutions of the above equation are the x-values of the intersection.
[tex]\begin{gathered} \cos2x=3\cos^2x-\sin^2x \\ \cos^2x-\sin^2x=3\cos^2x-\sin^2x \\ 2\cos^2x=0 \\ \cos^2x=0 \\ \cos x=0 \end{gathered}[/tex]
The solution to the above equation is:
[tex]x=\frac{\pi}{2}+2\pi n\text{ and }x=\frac{3\pi}{2}+2\pi n[/tex]
It is given that x lies between -pi and pi. So, the value of n = 0 for the first solution and n = 1 for the second solution. Therefore,
[tex]x=\frac{\pi}{2}\text{ and }x=-\frac{\pi}{2}[/tex]
Thus, options A and B are correct.
