If `|sin^(2)x+10x^(2)|=|9-x^(2)|+2sin^(2)x+cos^(2)x`, then x lies in

Options

A. `[-8. 8]`
B. `[-3, 3]`
C. `[-sqrt(17), sqrt(17)]`
D. `[-sqrt(21), sqrt(21)]`

Answer

To solve the equation \( | \sin^2 x + 10 - x^2 | = | 9 - x^2 | + 2 \sin^2 x + \cos^2 x \), we will follow these steps:

### Step 1: Rewrite the Equation
We start by rewriting the equation:
\[ | \sin^2 x + 10 - x^2 | = | 9 - x^2 | + 2 \sin^2 x + \cos^2 x \]
Since \( \cos^2 x = 1 - \sin^2 x \), we can substitute this into the equation:
\[ | \sin^2 x + 10 - x^2 | = | 9 - x^2 | + 2 \sin^2 x + (1 - \sin^2 x) \]
This simplifies to:
\[ | \sin^2 x + 10 - x^2 | = | 9 - x^2 | + \sin^2 x + 1 \]

### Step 2: Analyze the Absolute Values
Next, we need to consider the cases for the absolute values. We will analyze two cases based on the expressions inside the absolute values.

**Case 1:** \( \sin^2 x + 10 - x^2 \geq 0 \) and \( 9 - x^2 \geq 0 \)

In this case, we can drop the absolute values:
\[ \sin^2 x + 10 - x^2 = 9 - x^2 + \sin^2 x + 1 \]
This simplifies to:
\[ 10 = 10 \]
This is always true, so we need to find the conditions for which both expressions inside the absolute values are non-negative.

From \( 9 - x^2 \geq 0 \):
\[ x^2 \leq 9 \implies -3 \leq x \leq 3 \]

**Case 2:** \( \sin^2 x + 10 - x^2 < 0 \) and \( 9 - x^2 \geq 0 \)

Here, we have:
\[ -(\sin^2 x + 10 - x^2) = 9 - x^2 + \sin^2 x + 1 \]
This simplifies to:
\[ -x^2 - \sin^2 x - 10 = 9 - x^2 + \sin^2 x + 1 \]
Rearranging gives:
\[ -10 = 10 \sin^2 x + 10 \]
This leads to a contradiction, so there are no solutions from this case.

### Step 3: Conclusion
The only valid solutions come from Case 1, where \( -3 \leq x \leq 3 \).

Thus, the values of \( x \) that satisfy the original equation are:
\[ \boxed{[-3, 3]} \]