If `2 - cos^(2) theta = 3 sin theta cos theta , sin theta ne cos theta` then `tan theta` is

Options

A. `1/2`
B. `0`
C. `2/3`
D. `1/3`

Answer

To solve the equation \( 2 - \cos^2 \theta = 3 \sin \theta \cos \theta \) under the condition \( \sin \theta \neq \cos \theta \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 2 - \cos^2 \theta = 3 \sin \theta \cos \theta \]
We can rearrange this to:
\[ \cos^2 \theta + 3 \sin \theta \cos \theta - 2 = 0 \]

### Step 2: Use the Pythagorean identity
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can express \( \sin^2 \theta \) in terms of \( \cos^2 \theta \):
\[ \sin^2 \theta = 1 - \cos^2 \theta \]
Now, substituting \( \sin^2 \theta \) into the equation gives:
\[ 1 - \cos^2 \theta + 3 \sin \theta \cos \theta - 2 = 0 \]
This simplifies to:
\[ -\cos^2 \theta + 3 \sin \theta \cos \theta - 1 = 0 \]

### Step 3: Divide by \( \cos^2 \theta \)
Now, we divide the entire equation by \( \cos^2 \theta \):
\[ -\frac{\cos^2 \theta}{\cos^2 \theta} + \frac{3 \sin \theta \cos \theta}{\cos^2 \theta} - \frac{1}{\cos^2 \theta} = 0 \]
This simplifies to:
\[ -1 + 3 \tan \theta - \sec^2 \theta = 0 \]

### Step 4: Use the identity for \( \sec^2 \theta \)
Recall that \( \sec^2 \theta = 1 + \tan^2 \theta \). Substitute this into the equation:
\[ -1 + 3 \tan \theta - (1 + \tan^2 \theta) = 0 \]
This simplifies to:
\[ 3 \tan \theta - \tan^2 \theta - 2 = 0 \]

### Step 5: Rearranging the quadratic equation
Rearranging gives us:
\[ \tan^2 \theta - 3 \tan \theta + 2 = 0 \]

### Step 6: Factor the quadratic equation
Factoring the quadratic:
\[ (\tan \theta - 1)(\tan \theta - 2) = 0 \]
This gives us two possible solutions:
\[ \tan \theta = 1 \quad \text{or} \quad \tan \theta = 2 \]

### Step 7: Consider the condition \( \sin \theta \neq \cos \theta \)
Since we are given that \( \sin \theta \neq \cos \theta \), we cannot accept \( \tan \theta = 1 \) (which corresponds to \( \theta = 45^\circ \)). Therefore, we must have:
\[ \tan \theta = 2 \]

### Final Answer
Thus, the value of \( \tan \theta \) is:
\[ \boxed{2} \]