If ` sec x cos 5x=-1 and 0 lt x lt (pi)/(4)`, then x is equal to

Options

A. `(pi)/(6)`
B. `(pi)/(3)`
C. `(pi)/(4)`
D. None of these

Answer

To solve the equation \( \sec x \cos 5x = -1 \) for \( 0 < x < \frac{\pi}{4} \), we can follow these steps:

### Step 1: Rewrite the equation
The given equation is:
\[ \sec x \cos 5x = -1 \]
We know that \( \sec x = \frac{1}{\cos x} \), so we can rewrite the equation as:
\[ \frac{\cos 5x}{\cos x} = -1 \]

### Step 2: Rearranging the equation
Multiplying both sides by \( \cos x \) (noting that \( \cos x \neq 0 \) in the interval \( 0 < x < \frac{\pi}{4} \)):
\[ \cos 5x = -\cos x \]

### Step 3: Bringing terms to one side
Rearranging gives us:
\[ \cos x + \cos 5x = 0 \]

### Step 4: Using the cosine addition formula
We can use the formula for the sum of cosines:
\[ \cos a + \cos b = 2 \cos\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right) \]
Setting \( a = x \) and \( b = 5x \), we have:
\[ \cos x + \cos 5x = 2 \cos\left(\frac{x + 5x}{2}\right) \cos\left(\frac{x - 5x}{2}\right) = 2 \cos(3x) \cos(-2x) \]
Since \( \cos(-\theta) = \cos(\theta) \), this simplifies to:
\[ 2 \cos(3x) \cos(2x) = 0 \]

### Step 5: Setting each factor to zero
This gives us two equations to solve:
1. \( \cos(3x) = 0 \)
2. \( \cos(2x) = 0 \)

### Step 6: Solving \( \cos(3x) = 0 \)
The cosine function equals zero at odd multiples of \( \frac{\pi}{2} \):
\[ 3x = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]
For \( n = 0 \):
\[ 3x = \frac{\pi}{2} \implies x = \frac{\pi}{6} \]

### Step 7: Solving \( \cos(2x) = 0 \)
Similarly, for \( \cos(2x) = 0 \):
\[ 2x = \frac{\pi}{2} + m\pi \quad (m \in \mathbb{Z}) \]
For \( m = 0 \):
\[ 2x = \frac{\pi}{2} \implies x = \frac{\pi}{4} \]

### Step 8: Considering the interval
We need to check which solutions fall within the interval \( 0 < x < \frac{\pi}{4} \):
- From \( \cos(3x) = 0 \), we found \( x = \frac{\pi}{6} \) which is valid.
- From \( \cos(2x) = 0 \), we found \( x = \frac{\pi}{4} \) which is not valid since we need \( x < \frac{\pi}{4} \).

### Final Answer
Thus, the only solution in the interval \( 0 < x < \frac{\pi}{4} \) is:
\[ \boxed{\frac{\pi}{6}} \]