If ` tan theta = 1/sqrt7 " then" ((cosec^(2) theta -sec^(2) theta))/((cosec^(2)theta + sec^(2) theta))` is equal to

Options

A. `1/2`
B. `3/4`
C. `5/4`
D. 2

Explanation

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

**Given:**
\[ \tan \theta = \frac{1}{\sqrt{7}} \]

We need to find the value of:
\[ \frac{\csc^2 \theta - \sec^2 \theta}{\csc^2 \theta + \sec^2 \theta} \]

### Step 1: Use Trigonometric Identities
Recall the trigonometric identities:
\[ \csc^2 \theta = 1 + \cot^2 \theta \]
\[ \sec^2 \theta = 1 + \tan^2 \theta \]

### Step 2: Find \(\tan^2 \theta\)
From the given \(\tan \theta = \frac{1}{\sqrt{7}}\), we can find:
\[ \tan^2 \theta = \left(\frac{1}{\sqrt{7}}\right)^2 = \frac{1}{7} \]

### Step 3: Find \(\sec^2 \theta\)
Using the identity for \(\sec^2 \theta\):
\[ \sec^2 \theta = 1 + \tan^2 \theta = 1 + \frac{1}{7} = \frac{7}{7} + \frac{1}{7} = \frac{8}{7} \]

### Step 4: Find \(\cot^2 \theta\)
Using the identity \(\cot \theta = \frac{1}{\tan \theta}\):
\[ \cot \theta = \sqrt{7} \quad \Rightarrow \quad \cot^2 \theta = 7 \]

### Step 5: Find \(\csc^2 \theta\)
Using the identity for \(\csc^2 \theta\):
\[ \csc^2 \theta = 1 + \cot^2 \theta = 1 + 7 = 8 \]

### Step 6: Substitute Values into the Expression
Now we substitute \(\csc^2 \theta\) and \(\sec^2 \theta\) into the expression:
\[ \frac{\csc^2 \theta - \sec^2 \theta}{\csc^2 \theta + \sec^2 \theta} = \frac{8 - \frac{8}{7}}{8 + \frac{8}{7}} \]

### Step 7: Simplify the Numerator
To simplify the numerator:
\[ 8 - \frac{8}{7} = \frac{56}{7} - \frac{8}{7} = \frac{48}{7} \]

### Step 8: Simplify the Denominator
To simplify the denominator:
\[ 8 + \frac{8}{7} = \frac{56}{7} + \frac{8}{7} = \frac{64}{7} \]

### Step 9: Combine the Results
Now we have:
\[ \frac{\frac{48}{7}}{\frac{64}{7}} = \frac{48}{64} = \frac{3}{4} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{3}{4} \]