The value of x satisfying the equation `|sinx cosx|+sqrt(2+tan^(2)x+cot^(2)x)=sqrt3`

Options

A. `"belongs to "[0, (pi)/(3)]`
B. `"belongs to "((pi)/(3),(pi)/(2))`
C. `"belongs to "[(3pi)/(4),pi)`
D. does not exist

Answer

To solve the equation \( | \sin x \cos x | + \sqrt{2 + \tan^2 x + \cot^2 x} = \sqrt{3} \), we will follow these steps:

### Step 1: Simplify the equation
We start with the given equation:
\[ | \sin x \cos x | + \sqrt{2 + \tan^2 x + \cot^2 x} = \sqrt{3} \]

### Step 2: Rewrite \( \tan^2 x + \cot^2 x \)
Using the identity \( \tan^2 x + \cot^2 x = \frac{\sin^2 x}{\cos^2 x} + \frac{\cos^2 x}{\sin^2 x} \), we can rewrite it as:
\[ \tan^2 x + \cot^2 x = \frac{\sin^4 x + \cos^4 x}{\sin^2 x \cos^2 x} \]
However, we can also use the fact that \( \tan^2 x + \cot^2 x = \frac{\sin^2 x + \cos^2 x}{\sin^2 x \cos^2 x} - 2 \) to simplify our expression.

### Step 3: Substitute \( \tan^2 x + \cot^2 x \)
Thus, we can rewrite the square root term:
\[ \sqrt{2 + \tan^2 x + \cot^2 x} = \sqrt{2 + \frac{\sin^2 x + \cos^2 x}{\sin^2 x \cos^2 x} - 2} = \sqrt{\frac{1}{\sin^2 x \cos^2 x}} = \frac{1}{|\sin x \cos x|} \]

### Step 4: Substitute back into the equation
Now substituting this back into the original equation gives us:
\[ | \sin x \cos x | + \frac{1}{|\sin x \cos x|} = \sqrt{3} \]

### Step 5: Let \( y = |\sin x \cos x| \)
Let \( y = |\sin x \cos x| \). Then, we can rewrite the equation as:
\[ y + \frac{1}{y} = \sqrt{3} \]

### Step 6: Multiply through by \( y \)
Multiplying through by \( y \) (assuming \( y > 0 \)):
\[ y^2 - \sqrt{3}y + 1 = 0 \]

### Step 7: Solve the quadratic equation
Using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ y = \frac{\sqrt{3} \pm \sqrt{(\sqrt{3})^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{\sqrt{3} \pm \sqrt{-1}}{2} \]
Since the discriminant is negative, there are no real solutions for \( y \).

### Conclusion
Since \( y = |\sin x \cos x| \) cannot yield any real solutions, we conclude that there are no values of \( x \) that satisfy the original equation.

### Final Answer
Thus, the answer is that there are no values of \( x \) satisfying the equation.

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