If ` sec x cos 5x+1=0 `, where ` 0lt x lt 2pi `, then x=

Options

A. `(pi)/(5),(pi)/(4)`
B. `(pi)/(5)`
C. `(pi)/(4)`
D. None of these

Answer

To solve the equation \( \sec x \cos 5x + 1 = 0 \) for \( 0 < x < 2\pi \), we can follow these steps:

### Step 1: Rewrite the equation
Start by isolating the trigonometric terms:
\[ \sec x \cos 5x = -1 \]

### Step 2: Substitute secant
Recall that \( \sec x = \frac{1}{\cos x} \). Substitute this into the equation:
\[ \frac{\cos 5x}{\cos x} = -1 \]

### Step 3: Cross-multiply
Multiply both sides by \( \cos x \) (assuming \( \cos x \neq 0 \)):
\[ \cos 5x = -\cos x \]

### Step 4: Use the cosine identity
Using the identity \( \cos A = -\cos B \) gives us:
\[ 5x = (2n + 1)\pi - x \]
for some integer \( n \).

### Step 5: Rearrange the equation
Rearranging gives:
\[ 5x + x = (2n + 1)\pi \]
\[ 6x = (2n + 1)\pi \]
\[ x = \frac{(2n + 1)\pi}{6} \]

### Step 6: Find specific solutions
Now, we need to find values of \( n \) such that \( 0 < x < 2\pi \):
1. For \( n = 0 \):
\[ x = \frac{\pi}{6} \]
2. For \( n = 1 \):
\[ x = \frac{3\pi}{6} = \frac{\pi}{2} \]
3. For \( n = 2 \):
\[ x = \frac{5\pi}{6} \]
4. For \( n = 3 \):
\[ x = \frac{7\pi}{6} \]
5. For \( n = 4 \):
\[ x = \frac{9\pi}{6} = \frac{3\pi}{2} \]
6. For \( n = 5 \):
\[ x = \frac{11\pi}{6} \]

### Step 7: List the solutions
The solutions in the interval \( 0 < x < 2\pi \) are:
\[ x = \frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{3\pi}{2}, \frac{11\pi}{6} \]

### Step 8: Check for additional solutions
Now, we also need to consider the case where \( \cos 5x = \cos x \):
\[ 5x = 2n\pi \pm x \]
This leads to:
1. \( 6x = 2n\pi \) or \( 4x = 2n\pi \)

From \( 6x = 2n\pi \):
\[ x = \frac{n\pi}{3} \]
From \( 4x = 2n\pi \):
\[ x = \frac{n\pi}{2} \]

### Step 9: Find specific solutions from these cases
1. For \( n = 0, 1, 2, 3, 4, 5 \) in \( x = \frac{n\pi}{3} \):
- \( x = 0, \frac{\pi}{3}, \frac{2\pi}{3}, \pi, \frac{4\pi}{3}, \frac{5\pi}{3} \)
2. For \( n = 0, 1, 2, 3 \) in \( x = \frac{n\pi}{2} \):
- \( x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \)

### Final Step: Combine and filter solutions
Combining all valid solutions and filtering out those outside \( 0 < x < 2\pi \):
- Valid solutions are \( \frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{3\pi}{2}, \frac{11\pi}{6} \).

Among the options provided, the nearest solution that matches is:
\[ \text{Answer: } \frac{\pi}{4} \text{ (not listed, but } \frac{\pi}{2} \text{ is a valid solution)} \]