Find the value of x if `(4-sin^2 45^@)/(cot x. tan 60^@) `= `3.5`.

Options

A. `0^(@)`
B. `15^(@)`
C. `30^(@)`
D. `60^(@)`

Answer

To solve the equation \(\frac{4 - \sin^2 45^\circ}{\cot x \cdot \tan 60^\circ} = 3.5\), we will follow these steps:

### Step 1: Calculate \(\sin^2 45^\circ\)
We know that:
\[ \sin 45^\circ = \frac{1}{\sqrt{2}} \]
Thus,
\[ \sin^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

### Step 2: Substitute \(\sin^2 45^\circ\) into the equation
Now, substitute \(\sin^2 45^\circ\) into the equation:
\[ \frac{4 - \frac{1}{2}}{\cot x \cdot \tan 60^\circ} = 3.5 \]

### Step 3: Simplify the numerator
Calculate the numerator:
\[ 4 - \frac{1}{2} = \frac{8}{2} - \frac{1}{2} = \frac{7}{2} \]
So the equation becomes:
\[ \frac{\frac{7}{2}}{\cot x \cdot \tan 60^\circ} = 3.5 \]

### Step 4: Calculate \(\tan 60^\circ\)
We know:
\[ \tan 60^\circ = \sqrt{3} \]

### Step 5: Substitute \(\tan 60^\circ\) into the equation
Substituting \(\tan 60^\circ\) gives us:
\[ \frac{\frac{7}{2}}{\cot x \cdot \sqrt{3}} = 3.5 \]

### Step 6: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 7 = 3.5 \cdot \cot x \cdot \sqrt{3} \]

### Step 7: Solve for \(\cot x\)
Rearranging the equation:
\[ \cot x \cdot \sqrt{3} = \frac{7}{3.5} \]
Calculating \(\frac{7}{3.5}\):
\[ \frac{7}{3.5} = 2 \]
Thus:
\[ \cot x \cdot \sqrt{3} = 2 \]
Now, solving for \(\cot x\):
\[ \cot x = \frac{2}{\sqrt{3}} \]

### Step 8: Find \(x\)
To find \(x\), we take the cotangent inverse:
\[ x = \cot^{-1}\left(\frac{2}{\sqrt{3}}\right) \]

### Step 9: Convert to degrees
Using the cotangent value, we can find \(x\) in degrees. We know:
\[ \cot 30^\circ = \sqrt{3} \quad \text{and} \quad \cot 45^\circ = 1 \]
Since \(\frac{2}{\sqrt{3}}\) is between these two values, we can estimate \(x\) to be between \(30^\circ\) and \(45^\circ\).

### Final Answer
Thus, the value of \(x\) is:
\[ x \approx 30^\circ < x < 45^\circ \]