The number of solution s of the equation `tanx tan4 x = 1 " for" 0 lt x lt pi` is

Options

A. 1
B. 2
C. 5
D. 8

Answer

To solve the equation \( \tan x \tan 4x = 1 \) for \( 0 < x < \pi \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \tan x \tan 4x = 1 \]
We can rewrite this as:
\[ \tan 4x = \frac{1}{\tan x} \]
Using the identity \( \frac{1}{\tan x} = \cot x \), we can express this as:
\[ \tan 4x = \cot x \]

### Step 2: Use the cotangent identity
We know that:
\[ \cot x = \tan\left(\frac{\pi}{2} - x\right) \]
Thus, we can equate:
\[ \tan 4x = \tan\left(\frac{\pi}{2} - x\right) \]

### Step 3: Set up the general solution
From the property of the tangent function, if \( \tan A = \tan B \), then:
\[ A = n\pi + B \quad \text{for some integer } n \]
Applying this to our equation gives:
\[ 4x = n\pi + \left(\frac{\pi}{2} - x\right) \]

### Step 4: Solve for \( x \)
Rearranging the equation:
\[ 4x + x = n\pi + \frac{\pi}{2} \]
\[ 5x = n\pi + \frac{\pi}{2} \]
Now, solving for \( x \):
\[ x = \frac{n\pi}{5} + \frac{\pi}{10} \]

### Step 5: Determine valid values of \( n \)
We need to find the values of \( x \) that satisfy \( 0 < x < \pi \).

1. For \( n = 0 \):
\[ x = \frac{0\pi}{5} + \frac{\pi}{10} = \frac{\pi}{10} \]

2. For \( n = 1 \):
\[ x = \frac{1\pi}{5} + \frac{\pi}{10} = \frac{2\pi}{10} + \frac{\pi}{10} = \frac{3\pi}{10} \]

3. For \( n = 2 \):
\[ x = \frac{2\pi}{5} + \frac{\pi}{10} = \frac{4\pi}{10} + \frac{\pi}{10} = \frac{5\pi}{10} = \frac{\pi}{2} \]

4. For \( n = 3 \):
\[ x = \frac{3\pi}{5} + \frac{\pi}{10} = \frac{6\pi}{10} + \frac{\pi}{10} = \frac{7\pi}{10} \]

5. For \( n = 4 \):
\[ x = \frac{4\pi}{5} + \frac{\pi}{10} = \frac{8\pi}{10} + \frac{\pi}{10} = \frac{9\pi}{10} \]

6. For \( n = 5 \):
\[ x = \frac{5\pi}{5} + \frac{\pi}{10} = \pi + \frac{\pi}{10} > \pi \quad \text{(not valid)} \]

### Step 6: Count the valid solutions
The valid solutions for \( x \) in the interval \( (0, \pi) \) are:
1. \( \frac{\pi}{10} \)
2. \( \frac{3\pi}{10} \)
3. \( \frac{\pi}{2} \)
4. \( \frac{7\pi}{10} \)
5. \( \frac{9\pi}{10} \)

Thus, there are a total of **5 solutions**.

### Final Answer
The number of solutions of the equation \( \tan x \tan 4x = 1 \) for \( 0 < x < \pi \) is **5**.

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