If `7 sin^2 theta+3 cos^2 theta=4`, then find value of tan `theta`.

Answer

To solve the equation \( 7 \sin^2 \theta + 3 \cos^2 \theta = 4 \) and find the value of \( \tan \theta \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 7 \sin^2 \theta + 3 \cos^2 \theta = 4 \]

### Step 2: Use the Pythagorean identity
Recall the identity \( \sin^2 \theta + \cos^2 \theta = 1 \). We can express \( \cos^2 \theta \) in terms of \( \sin^2 \theta \):
\[ \cos^2 \theta = 1 - \sin^2 \theta \]
Substituting this into the original equation gives:
\[ 7 \sin^2 \theta + 3(1 - \sin^2 \theta) = 4 \]

### Step 3: Simplify the equation
Distributing the \( 3 \) in the equation:
\[ 7 \sin^2 \theta + 3 - 3 \sin^2 \theta = 4 \]
Combine like terms:
\[ (7 - 3) \sin^2 \theta + 3 = 4 \]
This simplifies to:
\[ 4 \sin^2 \theta + 3 = 4 \]

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

### Step 5: Find \( \cos^2 \theta \)
Using the identity again, we find \( \cos^2 \theta \):
\[ \cos^2 \theta = 1 - \sin^2 \theta = 1 - \frac{1}{4} = \frac{3}{4} \]

### Step 6: Find \( \tan^2 \theta \)
Now we can find \( \tan^2 \theta \):
\[ \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} = \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1}{3} \]

### Step 7: Find \( \tan \theta \)
Taking the square root gives:
\[ \tan \theta = \pm \sqrt{\frac{1}{3}} = \pm \frac{1}{\sqrt{3}} \]

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