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 of the equation \( \sin x \tan 4x = \cos x \) for \( x \in (0, \pi) \), we can follow these steps:

### 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: Use the Identity for Cotangent
Using the identity for cotangent, we can express it as:
\[ \tan 4x = \tan\left(\frac{\pi}{2} - x\right) \]

### Step 3: Set Up the General Solution
From the property of the tangent function, if \( \tan A = \tan B \), then:
\[ A = n\pi + B \]
where \( n \) is any integer. Applying this to our equation:
\[ 4x = n\pi + \left(\frac{\pi}{2} - x\right) \]

### Step 4: Solve for \( x \)
Rearranging gives:
\[ 4x + x = n\pi + \frac{\pi}{2} \]
\[ 5x = n\pi + \frac{\pi}{2} \]
\[ x = \frac{n\pi}{5} + \frac{\pi}{10} \]

### Step 5: Determine the Range of \( n \)
We need to find the values of \( n \) such that \( x \) remains in the interval \( (0, \pi) \):
\[ 0 < \frac{n\pi}{5} + \frac{\pi}{10} < \pi \]

### Step 6: Solve the Inequalities
1. For the left inequality:
\[ \frac{n\pi}{5} + \frac{\pi}{10} > 0 \]
This simplifies to:
\[ n\pi > -\frac{\pi}{10} \implies n > -\frac{1}{10} \]
Since \( n \) is an integer, the smallest value for \( n \) is \( 0 \).

2. For the right inequality:
\[ \frac{n\pi}{5} + \frac{\pi}{10} < \pi \]
This simplifies to:
\[ \frac{n\pi}{5} < \pi - \frac{\pi}{10} = \frac{9\pi}{10} \]
Dividing by \( \pi \) gives:
\[ \frac{n}{5} < \frac{9}{10} \implies n < \frac{9}{2} = 4.5 \]
Thus, the largest integer \( n \) can take is \( 4 \).

### Step 7: List Possible Values of \( n \)
The possible integer values for \( n \) are \( 0, 1, 2, 3, 4 \).

### Step 8: Count the Solutions
Thus, the total number of solutions for \( x \) in the interval \( (0, \pi) \) is:
\[ \text{Total Solutions} = 5 \]

### Final Answer
The total number of solutions of the equation \( \sin x \tan 4x = \cos x \) for \( x \in (0, \pi) \) is \( \boxed{5} \).

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