if `7 cos^(2) theta + 3 sin^2 theta = 4` and `0 lt theta lt (pi)/(2)`, then find the value of cot `theta`

Options

A. `sqrt(3)`
B. `3`
C. `1/3`
D. `1/(sqrt(3))`

Answer

To solve the equation \( 7 \cos^2 \theta + 3 \sin^2 \theta = 4 \) and find the value of \( \cot \theta \) for \( 0 < \theta < \frac{\pi}{2} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 7 \cos^2 \theta + 3 \sin^2 \theta = 4 \]
We can express \( \sin^2 \theta \) in terms of \( \cos^2 \theta \) using the identity \( \sin^2 \theta = 1 - \cos^2 \theta \).

### Step 2: Substitute \( \sin^2 \theta \)
Substituting \( \sin^2 \theta \) into the equation gives:
\[ 7 \cos^2 \theta + 3(1 - \cos^2 \theta) = 4 \]

### Step 3: Simplify the equation
Expanding the equation:
\[ 7 \cos^2 \theta + 3 - 3 \cos^2 \theta = 4 \]
Combine like terms:
\[ (7 - 3) \cos^2 \theta + 3 = 4 \]
\[ 4 \cos^2 \theta + 3 = 4 \]

### Step 4: Solve for \( \cos^2 \theta \)
Subtract 3 from both sides:
\[ 4 \cos^2 \theta = 4 - 3 \]
\[ 4 \cos^2 \theta = 1 \]
Now, divide by 4:
\[ \cos^2 \theta = \frac{1}{4} \]

### Step 5: Find \( \cos \theta \)
Taking the square root of both sides:
\[ \cos \theta = \frac{1}{2} \]
Since \( 0 < \theta < \frac{\pi}{2} \), we take the positive root:
\[ \cos \theta = \frac{1}{2} \]

### Step 6: Find \( \theta \)
We know that \( \cos 60^\circ = \frac{1}{2} \), thus:
\[ \theta = 60^\circ = \frac{\pi}{3} \]

### Step 7: Find \( \cot \theta \)
Using the definition of cotangent:
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]
We know \( \cos \theta = \frac{1}{2} \). To find \( \sin \theta \), we use the Pythagorean identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting \( \cos^2 \theta = \frac{1}{4} \):
\[ \sin^2 \theta + \frac{1}{4} = 1 \]
\[ \sin^2 \theta = 1 - \frac{1}{4} = \frac{3}{4} \]
Thus,
\[ \sin \theta = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

Now we can find \( \cot \theta \):
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \]

### Final Answer
Thus, the value of \( \cot \theta \) is:
\[ \cot \theta = \frac{1}{\sqrt{3}} \]