Blue line is graph `y=cos(3x)` and red is graph `y=cos x`. From graph you can see that intersection points are `x=0,\ pi/2,\ pi.` ` `
If you want to calculate those points you need to solve equation `cos3x=cosx.`
Let's first try to write `cos3x` by using only `cos x.` In order to do that we will use addition theorem:
`cos(t+s)=cos t cos s-sin t sin s`
and formula for cosine and sine of double angle:
` ` `cos2x=cos^2x-sin^2x`, `sin2x=2sinxcosx`.
`=cosx(cos^2x-sin^2x)-sin x cdot 2 sin x cos x=`
`=cos^3x-sin^2xcos x-2sin^2 x cos x=cos^3x-3sin^2x cos x=`
`=cos^3x-3(1-cos^2x)cos x=cos^3x -3cos x + 3cos^3x=`
Now we can solve our equation `cos3x=cosx.`
`4cos^3x-4cosx=0` devide equation by 4
`cosx(cos^2x-1)=0 =>` `cosx=0` or `cos^2-1=-sinx=0`
`cosx=0=>x=pi/2+k pi, k in ZZ` which is equal to `pi/2` for `x in [0,pi].`
`sinx=0=>x=k pi, k in ZZ` ` ` which is equal to 0 for `k=0` and `pi` for `k=1.`
So points of intersection are, as we concluded from the graph, `x=0,\ pi/2,\ pi.` If you want to find `y` coordinate of intersection points only calculate `cos x` for each of the points and get `y=1, 0, -1` respectively.