The number of distinct solution of `cos(x)/(4)=cos(x)` in `x in [0,24 pi]` is

Options

A. 23
B. 24
C. 25
D. 20

Answer

To solve the equation \(\frac{\cos(x)}{4} = \cos(x)\) for \(x\) in the interval \([0, 24\pi]\), we will follow these steps:

### Step 1: Rearranging the Equation
Start with the given equation:
\[ \frac{\cos(x)}{4} = \cos(x) \]
We can rearrange this to:
\[ \frac{\cos(x)}{4} - \cos(x) = 0 \]
This simplifies to:
\[ \frac{\cos(x) - 4\cos(x)}{4} = 0 \]
Which leads to:
\[ -3\cos(x) = 0 \]

### Step 2: Solving for \(\cos(x)\)
From the equation \(-3\cos(x) = 0\), we find:
\[ \cos(x) = 0 \]

### Step 3: Finding General Solutions
The general solution for \(\cos(x) = 0\) is given by:
\[ x = \frac{(2n + 1)\pi}{2} \]
where \(n\) is any integer.

### Step 4: Finding Specific Solutions in the Interval
Now we need to find the values of \(n\) such that \(x\) lies within the interval \([0, 24\pi]\):
\[ 0 \leq \frac{(2n + 1)\pi}{2} \leq 24\pi \]
Multiplying through by 2 gives:
\[ 0 \leq (2n + 1)\pi \leq 48\pi \]
Dividing by \(\pi\) (which is positive) results in:
\[ 0 \leq 2n + 1 \leq 48 \]
Subtracting 1 from all sides yields:
\[ -1 \leq 2n \leq 47 \]
Dividing by 2 gives:
\[ -\frac{1}{2} \leq n \leq 23.5 \]

### Step 5: Determining Integer Values for \(n\)
Since \(n\) must be an integer, the possible values for \(n\) are:
\[ n = 0, 1, 2, \ldots, 23 \]
This gives us a total of \(24\) distinct integer values for \(n\).

### Conclusion
Thus, the number of distinct solutions of \(\frac{\cos(x)}{4} = \cos(x)\) in the interval \([0, 24\pi]\) is:
\[ \boxed{24} \]