The variable `x` satisfying the equation `|sinxcosx|+sqrt(2+tan^2+cot^2x)=sqrt(3)` belongs to the interval `[0,pi/3]` (b) `(pi/3,pi/3)` (c) `[(3pi)/4,pi]` (d) none-existent

Options

A. `[0,pi/3]`
B. `(pi/3pi/2)`
C. `[(3pi)/4,pi)`
D. None of these

Explanation

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: Rewrite the Equation
We start with the equation:
\[ |\sin x \cos x| + \sqrt{2 + \tan^2 x + \cot^2 x} = \sqrt{3} \]

### Step 2: Simplify the Square Root Term
Recall that:
\[ \tan^2 x + \cot^2 x = \frac{\sin^2 x}{\cos^2 x} + \frac{\cos^2 x}{\sin^2 x} \]
Using the identity \( a + b \geq 2\sqrt{ab} \), we can simplify:
\[ \tan^2 x + \cot^2 x \geq 2 \]
Thus:
\[ 2 + \tan^2 x + \cot^2 x \geq 4 \]
And therefore:
\[ \sqrt{2 + \tan^2 x + \cot^2 x} \geq 2 \]

### Step 3: Analyze the Left Side of the Equation
Since \( |\sin x \cos x| \) is always non-negative, we have:
\[ |\sin x \cos x| + \sqrt{2 + \tan^2 x + \cot^2 x} \geq 2 + 0 = 2 \]

### Step 4: Compare with Right Side
The right side of the equation is \( \sqrt{3} \), which is approximately \( 1.732 \). Since the left side is always greater than or equal to \( 2 \), we can conclude:
\[ |\sin x \cos x| + \sqrt{2 + \tan^2 x + \cot^2 x} \geq 2 > \sqrt{3} \]

### Step 5: Conclusion
Since the left side is always greater than \( \sqrt{3} \), there are no values of \( x \) that satisfy the equation. Thus, the solution belongs to the option:
(d) none-existent.