### Instruction: Find the number of solutions to $\cos \frac{x}{4} = \cos x$ in the interval $0 < x < 24 \pi.$.### Response: I know that the cosine function is periodic with period $2 \pi,$ so it repeats itself every $2 \pi$ units along the x-axis.
This means that $\cos \frac{x}{4}$ and $\cos x$ will have the same value whenever $\frac{x}{4}$ and $x$ differ by a multiple of $2 \pi.$
In other words, I need to solve the equation $\frac{x}{4} - x = 2 \pi k,$ where $k$ is any integer.
I can simplify this equation by multiplying both sides by 4, which gives me $x - 4x = 8 \pi k,$ or $-3x = 8 \pi k.$
To isolate $x,$ I can divide both sides by -3, which gives me $x = -\frac{8}{3} \pi k.$
Now I need to find the values of $k$ that make $x$ lie in the given interval.
I can start by plugging in $k = 0,$ which gives me $x = 0.$
This is not in the interval, since I need $x > 0.$
So I can try $k = 1,$ which gives me $x = -\frac{8}{3} \pi.$
This is also not in the interval, since I need $x > 0.$
I can try $k = -1,$ which gives me $x = \frac{8}{3} \pi.$
This is in the interval, so it is one possible solution.
I can try $k = -2,$ which gives me $x = \frac{16}{3} \pi.$
This is also in the interval, so it is another possible solution.
I can try $k = -3,$ which gives me $x = 8 \pi.$
This is also in the interval, so it is another possible solution.
I can try $k = -4,$ which gives me $x = \frac{32}{3} \pi.$
This is also in the interval, so it is another possible solution.
I can try $k = -5,$ which gives me $x = \frac{40}{3} \pi.$
This is also in the interval, so it is another possible solution.
I can try $k = -6,$ which gives me $x = 16 \pi.$
This is also in the interval, so it is another possible solution.
I can try $k = -7,$ which gives me $x = \frac{56}{3} \pi.$
This is also in the interval, so it is another possible solution.
I can try $k = -8,$ which gives me $x = \frac{64}{3} \pi.$
This is also in the interval, so it is another possible solution.
I can try $k = -9,$ which gives me $x = 24 \pi.$
This is not in the interval, since I need $x < 24 \pi.$
So I have found all the possible values of $k$ that make $x$ a solution in the given interval.
There are 8 such values: $k = -1, -2, -3, -4, -5, -6, -7, -8.$
Therefore, there are 8 solutions to $\cos \frac{x}{4} = \cos x$ in the interval $0 < x < 24 \pi.$