If `y =4sin^(2)theta-cos2theta`,then `y` lies in the interval

Options

A. `(-1,5)`
B. `[-1,5]`
C. `(-oo,-1)uu(5,oo)`
D. none of these

Explanation

To solve the problem, we need to analyze the expression given for \( y \):

### Step-by-Step Solution:

1. **Write the given equation**:
\[ y = 4\sin^2\theta - \cos 2\theta \]

2. **Use the double angle identity for cosine**:
We know that:
\[ \cos 2\theta = 1 - 2\sin^2\theta \]
Therefore, we can rewrite \( y \) as:
\[ y = 4\sin^2\theta - (1 - 2\sin^2\theta) \]

3. **Simplify the expression**:
Distributing the negative sign:
\[ y = 4\sin^2\theta - 1 + 2\sin^2\theta \]
Combine like terms:
\[ y = (4\sin^2\theta + 2\sin^2\theta) - 1 = 6\sin^2\theta - 1 \]

4. **Determine the range of \( \sin^2\theta \)**:
The value of \( \sin^2\theta \) lies between 0 and 1:
\[ 0 \leq \sin^2\theta \leq 1 \]

5. **Substitute the bounds of \( \sin^2\theta \) into the equation for \( y \)**:
- When \( \sin^2\theta = 0 \):
\[ y = 6(0) - 1 = -1 \]
- When \( \sin^2\theta = 1 \):
\[ y = 6(1) - 1 = 5 \]

6. **Conclusion**:
Therefore, \( y \) lies in the interval:
\[ -1 \leq y \leq 5 \]

### Final Result:
Thus, the interval in which \( y \) lies is:
\[ [-1, 5] \]