If `7sin^(2)theta+3cos^(2)theta=4`, then value of `tantheta`.

Options

A. `(1)/(sqrt(3))`
B. `sqrt(3)`
C. `2sqrt(3)`
D. `(1)/(2sqrt(3))`

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

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

### Step 3: Isolate the cosine term
Rearranging the equation:
\[ -4\cos^2\theta = 4 - 7 \]
\[ -4\cos^2\theta = -3 \]
Dividing both sides by -4:
\[ \cos^2\theta = \frac{3}{4} \]

### Step 4: Solve for cosine
Taking the square root of both sides:
\[ \cos\theta = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2} \]

### Step 5: Find sine using the Pythagorean identity
Using \( \sin^2\theta + \cos^2\theta = 1 \):
\[ \sin^2\theta = 1 - \cos^2\theta = 1 - \frac{3}{4} = \frac{1}{4} \]
Taking the square root:
\[ \sin\theta = \pm \frac{1}{2} \]

### Step 6: Calculate \( \tan\theta \)
The tangent function is defined as:
\[ \tan\theta = \frac{\sin\theta}{\cos\theta} \]
Using the values we found:
1. If \( \cos\theta = \frac{\sqrt{3}}{2} \) and \( \sin\theta = \frac{1}{2} \):
\[ \tan\theta = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \]
2. If \( \cos\theta = -\frac{\sqrt{3}}{2} \) and \( \sin\theta = -\frac{1}{2} \):
\[ \tan\theta = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \]

Thus, in both cases, the value of \( \tan\theta \) is:
\[ \tan\theta = \frac{1}{\sqrt{3}} \quad \text{or} \quad \tan\theta = \frac{\sqrt{3}}{3} \]

### Final Answer
The value of \( \tan\theta \) is \( \frac{1}{\sqrt{3}} \) or \( \frac{\sqrt{3}}{3} \).

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