Find the number of real solution of the equation `(cos x)^(5)+(sin x)^(3)=1` in the interval `[0, 2pi]`

Explanation

To find the number of real solutions for the equation \((\cos x)^5 + (\sin x)^3 = 1\) in the interval \([0, 2\pi]\), we can follow these steps:

### Step 1: Analyze the equation
The equation is given as:
\[ (\cos x)^5 + (\sin x)^3 = 1 \]
We know that both \((\cos x)^5\) and \((\sin x)^3\) are non-negative for values of \(x\) in the interval \([0, 2\pi]\).

### Step 2: Identify the maximum values
The maximum value of \((\cos x)^5\) occurs when \(\cos x = 1\), which gives \((\cos x)^5 = 1\). The maximum value of \((\sin x)^3\) occurs when \(\sin x = 1\), which gives \((\sin x)^3 = 1\). Therefore, the maximum sum of \((\cos x)^5 + (\sin x)^3\) can be at most 1.

### Step 3: Set conditions for the equation to hold
For the equation \((\cos x)^5 + (\sin x)^3 = 1\) to hold, we can consider two cases:
1. \((\cos x)^5 = 1\) and \((\sin x)^3 = 0\)
2. \((\cos x)^5 < 1\) and \((\sin x)^3 > 0\)

### Step 4: Solve for the first case
For the first case:
- \((\cos x)^5 = 1\) implies \(\cos x = 1\). This occurs at:
- \(x = 0\)
- \(x = 2\pi\)

- \((\sin x)^3 = 0\) implies \(\sin x = 0\). This occurs at:
- \(x = 0\)
- \(x = \pi\)
- \(x = 2\pi\)

From this case, we have the solutions \(x = 0\) and \(x = 2\pi\).

### Step 5: Solve for the second case
For the second case:
- We need to find values of \(x\) where \((\cos x)^5 + (\sin x)^3 = 1\) holds true but neither term is equal to 1 exclusively.

This can occur when:
- \(\cos x\) is less than 1 (which means \(\sin x\) must be greater than 0, thus \(0 < x < \pi\)).
- We need to check specific angles within this range.

### Step 6: Check specific angles
1. At \(x = \frac{\pi}{2}\):
- \(\cos\left(\frac{\pi}{2}\right) = 0\)
- \(\sin\left(\frac{\pi}{2}\right) = 1\)
- Thus, \((\cos x)^5 + (\sin x)^3 = 0 + 1 = 1\) (valid solution).

### Step 7: Compile the solutions
From our analysis, we found three solutions:
- \(x = 0\)
- \(x = \frac{\pi}{2}\)
- \(x = 2\pi\)

### Conclusion
The total number of real solutions in the interval \([0, 2\pi]\) is **3**.