If `(cos^(2) theta)/(cot^(2) theta - cos^(2) theta) = 3` and `0^(@) lt theta lt 90^(@)` , then the value of `theta` is

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. None of these

Answer

To solve the equation \(\frac{\cos^2 \theta}{\cot^2 \theta - \cos^2 \theta} = 3\), we will follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ \frac{\cos^2 \theta}{\cot^2 \theta - \cos^2 \theta} = 3 \]
Multiply both sides by \((\cot^2 \theta - \cos^2 \theta)\) to eliminate the fraction:
\[ \cos^2 \theta = 3(\cot^2 \theta - \cos^2 \theta) \]

### Step 2: Expand the equation
Distributing the 3 on the right side gives:
\[ \cos^2 \theta = 3\cot^2 \theta - 3\cos^2 \theta \]

### Step 3: Rearrange the equation
Bring all terms involving \(\cos^2 \theta\) to one side:
\[ \cos^2 \theta + 3\cos^2 \theta = 3\cot^2 \theta \]
This simplifies to:
\[ 4\cos^2 \theta = 3\cot^2 \theta \]

### Step 4: Substitute \(\cot \theta\)
Recall that \(\cot \theta = \frac{\cos \theta}{\sin \theta}\), so:
\[ \cot^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta} \]
Substituting this into the equation gives:
\[ 4\cos^2 \theta = 3 \cdot \frac{\cos^2 \theta}{\sin^2 \theta} \]

### Step 5: Simplify the equation
Multiply both sides by \(\sin^2 \theta\) to eliminate the fraction:
\[ 4\cos^2 \theta \sin^2 \theta = 3\cos^2 \theta \]
Assuming \(\cos^2 \theta \neq 0\), we can divide both sides by \(\cos^2 \theta\):
\[ 4\sin^2 \theta = 3 \]

### Step 6: Solve for \(\sin^2 \theta\)
Rearranging gives:
\[ \sin^2 \theta = \frac{3}{4} \]

### Step 7: Take the square root
Taking the square root of both sides, we find:
\[ \sin \theta = \frac{\sqrt{3}}{2} \]

### Step 8: Find \(\theta\)
The angle \(\theta\) that satisfies \(\sin \theta = \frac{\sqrt{3}}{2}\) in the interval \(0^\circ < \theta < 90^\circ\) is:
\[ \theta = 60^\circ \]

Thus, the value of \(\theta\) is \(60^\circ\).

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