Number of solution (s) of equation ` sin theta = sec^(2) 4 theta ` in `[0, pi]` is/are :

Options

A. 0
B. 1
C. 2
D. 3

Answer

To find the number of solutions of the equation \( \sin \theta = \sec^2(4\theta) \) in the interval \( [0, \pi] \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \sin \theta = \sec^2(4\theta) \]
Since \( \sec^2(4\theta) = \frac{1}{\cos^2(4\theta)} \), we can rewrite the equation as:
\[ \sin \theta = \frac{1}{\cos^2(4\theta)} \]

### Step 2: Cross-multiply
Cross-multiplying gives us:
\[ \sin \theta \cdot \cos^2(4\theta) = 1 \]

### Step 3: Analyze the equation
Now we have the equation:
\[ \sin \theta \cdot \cos^2(4\theta) = 1 \]
This implies that \( \sin \theta \) must be positive and \( \cos^2(4\theta) \) must also be positive. The maximum value of \( \sin \theta \) is 1, which occurs at \( \theta = \frac{\pi}{2} \). Thus, we need to check when \( \sin \theta \) can equal 1.

### Step 4: Set up conditions
1. **Condition for \( \sin \theta = 1 \)**:
\[ \sin \theta = 1 \implies \theta = \frac{\pi}{2} \]

2. **Condition for \( \cos^2(4\theta) = 1 \)**:
\[ \cos^2(4\theta) = 1 \implies \cos(4\theta) = \pm 1 \]
This occurs when:
\[ 4\theta = n\pi \quad \text{for } n \in \mathbb{Z} \]
Thus,
\[ \theta = \frac{n\pi}{4} \]

### Step 5: Find values of \( \theta \) in the interval \( [0, \pi] \)
We need to find the values of \( n \) such that \( \frac{n\pi}{4} \) lies in the interval \( [0, \pi] \):
- For \( n = 0 \), \( \theta = 0 \)
- For \( n = 1 \), \( \theta = \frac{\pi}{4} \)
- For \( n = 2 \), \( \theta = \frac{\pi}{2} \)
- For \( n = 3 \), \( \theta = \frac{3\pi}{4} \)
- For \( n = 4 \), \( \theta = \pi \)

Thus, the possible values of \( \theta \) are:
\[ 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi \]

### Step 6: Check which values satisfy the original equation
Now we need to check which of these values satisfy the original equation \( \sin \theta \cdot \cos^2(4\theta) = 1 \):
- For \( \theta = 0 \): \( \sin(0) \cdot \cos^2(0) = 0 \) (not a solution)
- For \( \theta = \frac{\pi}{4} \): \( \sin\left(\frac{\pi}{4}\right) \cdot \cos^2(\pi) = \frac{\sqrt{2}}{2} \cdot 0 = 0 \) (not a solution)
- For \( \theta = \frac{\pi}{2} \): \( \sin\left(\frac{\pi}{2}\right) \cdot \cos^2(2\pi) = 1 \cdot 1 = 1 \) (solution)
- For \( \theta = \frac{3\pi}{4} \): \( \sin\left(\frac{3\pi}{4}\right) \cdot \cos^2(3\pi) = \frac{\sqrt{2}}{2} \cdot 0 = 0 \) (not a solution)
- For \( \theta = \pi \): \( \sin(\pi) \cdot \cos^2(4\pi) = 0 \cdot 1 = 0 \) (not a solution)

### Conclusion
The only value of \( \theta \) that satisfies the equation in the interval \( [0, \pi] \) is \( \frac{\pi}{2} \).

Thus, the number of solutions is:
\[ \boxed{1} \]