If `y= sin^(2)theta + cosec^(2)theta, theta ne 0`, then

Options

A. y=0
B. `y le 2`
C. `y ge -2`
D. `y ge 2`

Answer

To solve the problem where \( y = \sin^2 \theta + \csc^2 \theta \) and \( \theta \neq 0 \), we will follow these steps:

### Step-by-Step Solution:

1. **Start with the given expression:**
\[ y = \sin^2 \theta + \csc^2 \theta \]

2. **Recall the identity for cosecant:**
\[ \csc \theta = \frac{1}{\sin \theta} \]
Therefore, we can express \( \csc^2 \theta \) as:
\[ \csc^2 \theta = \frac{1}{\sin^2 \theta} \]

3. **Substitute \( \csc^2 \theta \) into the expression for \( y \):**
\[ y = \sin^2 \theta + \frac{1}{\sin^2 \theta} \]

4. **Let \( x = \sin^2 \theta \). Then we can rewrite \( y \) as:**
\[ y = x + \frac{1}{x} \]
where \( x > 0 \) since \( \sin^2 \theta \) is always positive for \( \theta \neq 0 \).

5. **To find the minimum value of \( y \), we can use the AM-GM inequality:**
\[ x + \frac{1}{x} \geq 2 \]
This inequality holds for all \( x > 0 \).

6. **Equality holds when \( x = 1 \) (or \( \sin^2 \theta = 1 \), which occurs when \( \theta = \frac{\pi}{2} + n\pi \) for any integer \( n \)).**

7. **Thus, the minimum value of \( y \) is:**
\[ y \geq 2 \]

8. **Conclusion:**
The minimum value of \( y \) is 2, and it can be expressed as:
\[ y \geq 2 \]

### Final Answer:
\[ y \geq 2 \]