If the equation `1 +sin^2 xtheta= costheta` has a non-zero solution in `theta`, then `x` must be (1) An integer (2) A rational number (3) An irrational number (4) None of these

Options

A. An integer
B. A rational number
C. An irrational number
D. None of these

Explanation

To solve the equation \(1 + \sin^2(x\theta) = \cos(\theta)\) and determine the nature of \(x\) when there is a non-zero solution in \(\theta\), we can follow these steps:

### Step 1: Analyze the given equation
We start with the equation:
\[ 1 + \sin^2(x\theta) = \cos(\theta) \]

### Step 2: Determine the range of \(\sin^2(x\theta)\)
We know that \(\sin^2(x\theta)\) varies from 0 to 1 because the sine function ranges from -1 to 1. Therefore:
\[ 0 \leq \sin^2(x\theta) \leq 1 \]
Adding 1 to all sides gives:
\[ 1 \leq 1 + \sin^2(x\theta) \leq 2 \]

### Step 3: Determine the range of \(\cos(\theta)\)
The cosine function also has a specific range:
\[ -1 \leq \cos(\theta) \leq 1 \]

### Step 4: Compare the ranges
From Step 2, we have:
\[ 1 \leq 1 + \sin^2(x\theta) \leq 2 \]
From Step 3, we have:
\[ -1 \leq \cos(\theta) \leq 1 \]
For the equation \(1 + \sin^2(x\theta) = \cos(\theta)\) to hold, both sides must be equal. Therefore, we can conclude:
\[ 1 \leq \cos(\theta) \leq 1 \]
This implies:
\[ \cos(\theta) = 1 \]

### Step 5: Solve for \(\theta\)
The condition \(\cos(\theta) = 1\) occurs when:
\[ \theta = 2n\pi \quad \text{for } n \in \mathbb{Z} \]

### Step 6: Substitute \(\theta\) back into the equation
Substituting \(\theta = 2n\pi\) into the original equation gives:
\[ 1 + \sin^2(x(2n\pi)) = 1 \]
This simplifies to:
\[ \sin^2(x(2n\pi)) = 0 \]
This means:
\[ \sin(x(2n\pi)) = 0 \]

### Step 7: Solve for \(x\)
The sine function is zero at integer multiples of \(\pi\):
\[ x(2n\pi) = k\pi \quad \text{for } k \in \mathbb{Z} \]
Dividing both sides by \(\pi\) gives:
\[ 2nx = k \]
Thus:
\[ x = \frac{k}{2n} \]
Since \(n\) can be any non-zero integer, \(x\) can take values that are rational numbers (as \(k\) and \(n\) are integers).

### Conclusion
Since \(x\) can be expressed as a ratio of integers, we conclude that \(x\) must be a rational number.

### Final Answer
The correct option is:
(2) A rational number.