If `sec^(2)theta=3, 0^(@)ltthetalt(pi)/(2)`, then the value of `(tan^(2)theta-cosec^(2)theta)/(tan^(2)theta+cosec^(2)theta)` is

Options

A. `(4)/(7)`
B. `(2)/(7)`
C. `(1)/(7)`
D. `(3)/(7)`

Answer

To solve the problem, we start with the given information:

1. \( \sec^2 \theta = 3 \)
2. We need to find the value of \( \frac{\tan^2 \theta - \csc^2 \theta}{\tan^2 \theta + \csc^2 \theta} \)

### Step 1: Find \( \sec \theta \)

Since \( \sec^2 \theta = 3 \), we can find \( \sec \theta \):

\[ \sec \theta = \sqrt{3} \]

### Step 2: Relate \( \sec \theta \) to the sides of a triangle

Recall that \( \sec \theta = \frac{\text{hypotenuse}}{\text{base}} \). We can set up a right triangle where:

- Hypotenuse = \( \sqrt{3} \)
- Base = 1

To find the height (opposite side), we use the Pythagorean theorem:

\[ \text{height} = \sqrt{(\text{hypotenuse})^2 - (\text{base})^2} = \sqrt{(\sqrt{3})^2 - 1^2} = \sqrt{3 - 1} = \sqrt{2} \]

### Step 3: Find \( \tan \theta \) and \( \cos \theta \)

Now we can find \( \tan \theta \) and \( \cos \theta \):

\[ \tan \theta = \frac{\text{height}}{\text{base}} = \frac{\sqrt{2}}{1} = \sqrt{2} \]

\[ \cos \theta = \frac{\text{base}}{\text{hypotenuse}} = \frac{1}{\sqrt{3}} \]

### Step 4: Calculate \( \tan^2 \theta \) and \( \csc^2 \theta \)

Now we calculate \( \tan^2 \theta \) and \( \csc^2 \theta \):

\[ \tan^2 \theta = (\sqrt{2})^2 = 2 \]

\[ \csc^2 \theta = \frac{1}{\sin^2 \theta} = 1 + \cot^2 \theta = 1 + \frac{1}{\tan^2 \theta} = 1 + \frac{1}{2} = \frac{3}{2} \]

### Step 5: Substitute into the expression

Now substitute \( \tan^2 \theta \) and \( \csc^2 \theta \) into the expression:

\[ \frac{\tan^2 \theta - \csc^2 \theta}{\tan^2 \theta + \csc^2 \theta} = \frac{2 - \frac{3}{2}}{2 + \frac{3}{2}} \]

### Step 6: Simplify the expression

Calculating the numerator:

\[ 2 - \frac{3}{2} = \frac{4}{2} - \frac{3}{2} = \frac{1}{2} \]

Calculating the denominator:

\[ 2 + \frac{3}{2} = \frac{4}{2} + \frac{3}{2} = \frac{7}{2} \]

### Step 7: Final calculation

Now we have:

\[ \frac{\frac{1}{2}}{\frac{7}{2}} = \frac{1}{2} \cdot \frac{2}{7} = \frac{1}{7} \]

Thus, the final answer is:

\[ \frac{1}{7} \]