If` P = 2sin^(2)x - cos^(2)x,` then P lies in the interval

Options

A. [1, 3]
B. [1, 2]
C. [–1, 2]
D. None of these

Answer

To find the interval in which \( P = 2\sin^2 x - \cos^2 x \) lies, we can follow these steps:

### Step 1: Rewrite the expression for \( P \)
We know that \( \cos^2 x = 1 - \sin^2 x \). Therefore, we can substitute this into the expression for \( P \):

\[ P = 2\sin^2 x - \cos^2 x = 2\sin^2 x - (1 - \sin^2 x) \]

### Step 2: Simplify the expression
Now, we can simplify the expression:

\[ P = 2\sin^2 x - 1 + \sin^2 x = 3\sin^2 x - 1 \]

### Step 3: Determine the range of \( \sin^2 x \)
The function \( \sin^2 x \) varies between 0 and 1 for all \( x \):

\[ 0 \leq \sin^2 x \leq 1 \]

### Step 4: Substitute the range of \( \sin^2 x \) into \( P \)
Now, we can substitute the minimum and maximum values of \( \sin^2 x \) into the expression for \( P \):

1. When \( \sin^2 x = 0 \):
\[ P = 3(0) - 1 = -1 \]

2. When \( \sin^2 x = 1 \):
\[ P = 3(1) - 1 = 2 \]

### Step 5: Conclusion
Thus, the values of \( P \) will range from -1 to 2:

\[ P \in [-1, 2] \]

### Final Answer
The interval in which \( P \) lies is \([-1, 2]\).

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