If `3(cot^(2)theta-cos^(2)theta)=1-sin^(2)theta,0^(@)ltthetalt90^(@),` then the `theta` equal to

Options

A. A) `30^(@)`
B. B) `60^(@)`
C. C) `45^(@)`
D. D) `15^(@)`

Answer

To solve the equation \(3(\cot^2 \theta - \cos^2 \theta) = 1 - \sin^2 \theta\) for \(0^\circ < \theta < 90^\circ\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ 3(\cot^2 \theta - \cos^2 \theta) = 1 - \sin^2 \theta \]

### Step 2: Use the Pythagorean identity
Recall the Pythagorean identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
From this, we can express \(1 - \sin^2 \theta\) as:
\[ 1 - \sin^2 \theta = \cos^2 \theta \]

### Step 3: Substitute into the equation
Substituting \(1 - \sin^2 \theta\) into the equation gives:
\[ 3(\cot^2 \theta - \cos^2 \theta) = \cos^2 \theta \]

### Step 4: Simplify the equation
Now, we can simplify the equation:
\[ 3\cot^2 \theta - 3\cos^2 \theta = \cos^2 \theta \]
Rearranging terms yields:
\[ 3\cot^2 \theta = 4\cos^2 \theta \]

### Step 5: Express \(\cot^2 \theta\) in terms of \(\sin^2 \theta\)
Recall that:
\[ \cot^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta} \]
Substituting this into the equation gives:
\[ 3\left(\frac{\cos^2 \theta}{\sin^2 \theta}\right) = 4\cos^2 \theta \]

### Step 6: Cross-multiply to eliminate the fraction
Cross-multiplying results in:
\[ 3\cos^2 \theta = 4\cos^2 \theta \sin^2 \theta \]

### Step 7: Rearranging the equation
Rearranging gives:
\[ 4\cos^2 \theta \sin^2 \theta - 3\cos^2 \theta = 0 \]
Factoring out \(\cos^2 \theta\) gives:
\[ \cos^2 \theta (4\sin^2 \theta - 3) = 0 \]

### Step 8: Solve for \(\theta\)
Since \(\cos^2 \theta = 0\) would imply \(\theta = 90^\circ\), which is not in our interval, we set:
\[ 4\sin^2 \theta - 3 = 0 \]
Solving for \(\sin^2 \theta\) gives:
\[ \sin^2 \theta = \frac{3}{4} \]
Taking the square root:
\[ \sin \theta = \frac{\sqrt{3}}{2} \]

### Step 9: Find the angle \(\theta\)
The angle \(\theta\) for which \(\sin \theta = \frac{\sqrt{3}}{2}\) in the interval \(0^\circ < \theta < 90^\circ\) is:
\[ \theta = 60^\circ \]

### Final Answer
Thus, the value of \(\theta\) is:
\[ \theta = 60^\circ \]