The total number of solutions of the equation `sinx tan4x=cosx` for all `x in (0, pi)` are

Answer

To find the total number of solutions for the equation \( \sin x \tan 4x = \cos x \) in the interval \( (0, \pi) \), we can manipulate the equation and analyze the graphs of the functions involved. Here is a step-by-step solution:

### Step 1: Rewrite the Equation
We start with the equation:
\[ \sin x \tan 4x = \cos x \]
We can rewrite this as:
\[ \tan 4x = \frac{\cos x}{\sin x} = \cot x \]

### Step 2: Analyze the Functions
Now we need to analyze the functions \( y = \tan 4x \) and \( y = \cot x \) to find their points of intersection.

### Step 3: Determine the Period of the Functions
The period of \( \tan x \) is \( \pi \). Therefore, the period of \( \tan 4x \) is:
\[ \text{Period of } \tan 4x = \frac{\pi}{4} \]
This means that \( \tan 4x \) will complete one full cycle in \( \frac{\pi}{4} \).

### Step 4: Sketch the Graphs
1. **Graph of \( y = \cot x \)**:
- The graph of \( \cot x \) has vertical asymptotes at \( x = 0 \) and \( x = \pi \).
- It decreases from \( +\infty \) at \( x = 0 \) to \( -\infty \) at \( x = \pi \), crossing the x-axis at \( x = \frac{\pi}{2} \).

2. **Graph of \( y = \tan 4x \)**:
- The graph of \( \tan 4x \) has vertical asymptotes at \( x = \frac{\pi}{8}, \frac{3\pi}{8}, \frac{5\pi}{8}, \frac{7\pi}{8} \) within the interval \( (0, \pi) \).
- It oscillates between \( -\infty \) and \( +\infty \) within each period of \( \frac{\pi}{4} \).

### Step 5: Find Points of Intersection
To find the total number of solutions, we need to count how many times the graphs of \( y = \tan 4x \) and \( y = \cot x \) intersect in the interval \( (0, \pi) \).

- Between each pair of asymptotes of \( \tan 4x \) (which occur every \( \frac{\pi}{8} \)), there is one intersection with \( \cot x \).
- The intervals where intersections occur are:
- \( (0, \frac{\pi}{8}) \)
- \( (\frac{\pi}{8}, \frac{3\pi}{8}) \)
- \( (\frac{3\pi}{8}, \frac{5\pi}{8}) \)
- \( (\frac{5\pi}{8}, \frac{7\pi}{8}) \)
- \( (\frac{7\pi}{8}, \pi) \)

### Step 6: Count the Intersections
In total, we find that there are 5 points of intersection between the graphs of \( y = \tan 4x \) and \( y = \cot x \) in the interval \( (0, \pi) \).

### Conclusion
Thus, the total number of solutions to the equation \( \sin x \tan 4x = \cos x \) for \( x \in (0, \pi) \) is:
\[ \boxed{5} \]