If `cot theta = (7)/(8)`, then the value of `((1 + sin theta)(1- sin theta))/((1 + cos theta)(1- cos theta))` = _________

Answer

To solve the problem, we need to find the value of the expression:

\[ \frac{(1 + \sin \theta)(1 - \sin \theta)}{(1 + \cos \theta)(1 - \cos \theta)} \]

Given that \( \cot \theta = \frac{7}{8} \).

### Step 1: Use the cotangent identity
Since \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), we can express \( \cos \theta \) and \( \sin \theta \) in terms of a right triangle. Let:
- Opposite side = 7 (for sine)
- Adjacent side = 8 (for cosine)

Using the Pythagorean theorem, we can find the hypotenuse \( r \):
\[ r = \sqrt{7^2 + 8^2} = \sqrt{49 + 64} = \sqrt{113} \]

Thus, we have:
\[ \sin \theta = \frac{7}{\sqrt{113}}, \quad \cos \theta = \frac{8}{\sqrt{113}} \]

### Step 2: Substitute into the expression
Now we substitute \( \sin \theta \) and \( \cos \theta \) into the expression:

\[ 1 + \sin \theta = 1 + \frac{7}{\sqrt{113}} = \frac{\sqrt{113} + 7}{\sqrt{113}} \]
\[ 1 - \sin \theta = 1 - \frac{7}{\sqrt{113}} = \frac{\sqrt{113} - 7}{\sqrt{113}} \]
\[ 1 + \cos \theta = 1 + \frac{8}{\sqrt{113}} = \frac{\sqrt{113} + 8}{\sqrt{113}} \]
\[ 1 - \cos \theta = 1 - \frac{8}{\sqrt{113}} = \frac{\sqrt{113} - 8}{\sqrt{113}} \]

### Step 3: Calculate the numerator and denominator
Now we can calculate the numerator and denominator of the expression:

**Numerator:**
\[ (1 + \sin \theta)(1 - \sin \theta) = \left(\frac{\sqrt{113} + 7}{\sqrt{113}}\right)\left(\frac{\sqrt{113} - 7}{\sqrt{113}}\right) = \frac{(\sqrt{113})^2 - 7^2}{113} = \frac{113 - 49}{113} = \frac{64}{113} \]

**Denominator:**
\[ (1 + \cos \theta)(1 - \cos \theta) = \left(\frac{\sqrt{113} + 8}{\sqrt{113}}\right)\left(\frac{\sqrt{113} - 8}{\sqrt{113}}\right) = \frac{(\sqrt{113})^2 - 8^2}{113} = \frac{113 - 64}{113} = \frac{49}{113} \]

### Step 4: Combine the results
Now we can combine the results to find the value of the expression:

\[ \frac{(1 + \sin \theta)(1 - \sin \theta)}{(1 + \cos \theta)(1 - \cos \theta)} = \frac{\frac{64}{113}}{\frac{49}{113}} = \frac{64}{49} \]

### Final Answer
Thus, the value of the expression is:

\[ \frac{64}{49} \]