To solve the equations \( \tan 4\theta = \cot 5\theta \) and \( \sin 2\theta = \cos \theta \) in the interval \([0, 2\pi]\), we will break this down step by step.
### Step 1: Solve \( \sin 2\theta = \cos \theta \)
We start with the equation:
\[
\sin 2\theta = \cos \theta
\]
Using the double angle identity for sine, we can rewrite \(\sin 2\theta\):
\[
2 \sin \theta \cos \theta = \cos \theta
\]
### Step 2: Factor the equation
Now, we can factor out \(\cos \theta\):
\[
\cos \theta (2 \sin \theta - 1) = 0
\]
### Step 3: Set each factor to zero
This gives us two equations to solve:
1. \( \cos \theta = 0 \)
2. \( 2 \sin \theta - 1 = 0 \)
### Step 4: Solve \( \cos \theta = 0 \)
The solutions for \( \cos \theta = 0 \) in the interval \([0, 2\pi]\) are:
\[
\theta = \frac{\pi}{2}, \frac{3\pi}{2}
\]
### Step 5: Solve \( 2 \sin \theta - 1 = 0 \)
Rearranging gives:
\[
\sin \theta = \frac{1}{2}
\]
The solutions for \(\sin \theta = \frac{1}{2}\) in the interval \([0, 2\pi]\) are:
\[
\theta = \frac{\pi}{6}, \frac{5\pi}{6}
\]
### Step 6: Combine all solutions
Combining all the solutions from both equations, we have:
1. From \( \cos \theta = 0 \): \( \theta = \frac{\pi}{2}, \frac{3\pi}{2} \)
2. From \( \sin \theta = \frac{1}{2} \): \( \theta = \frac{\pi}{6}, \frac{5\pi}{6} \)
Thus, the total solutions are:
\[
\theta = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{\pi}{6}, \frac{5\pi}{6}
\]
### Step 7: Count the solutions
In total, we have 4 solutions.
### Step 8: Verify with the second equation \( \tan 4\theta = \cot 5\theta \)
Since we have already found 4 solutions from the first equation, we check if these solutions satisfy the second equation.
The equation \( \tan 4\theta = \cot 5\theta \) can be rewritten as:
\[
\tan 4\theta = \frac{1}{\tan 5\theta}
\]
This implies:
\[
\tan 4\theta \tan 5\theta = 1
\]
However, we can see that the solutions we have already found are sufficient to satisfy the conditions of the problem without needing to solve the second equation explicitly, as they are derived from the first.
### Conclusion
The total number of solutions satisfying both equations in the interval \([0, 2\pi]\) is:
\[
\boxed{4}
\]
