If `cot theta= (7)/(8)` then the value of `((1+ sintheta)(1-sintheta))/((1+costheta)(1-costheta))`= …………….

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-by-Step Solution:

1. **Use the identity for the numerator and denominator**:
The numerator can be simplified using the identity \( (1 + \sin \theta)(1 - \sin \theta) = 1 - \sin^2 \theta \).
Similarly, the denominator can be simplified as \( (1 + \cos \theta)(1 - \cos \theta) = 1 - \cos^2 \theta \).

Therefore, we can rewrite the expression as:
\[ \frac{1 - \sin^2 \theta}{1 - \cos^2 \theta} \]

2. **Apply the Pythagorean identity**:
According to the Pythagorean identity, we know that:
\[ 1 - \sin^2 \theta = \cos^2 \theta \quad \text{and} \quad 1 - \cos^2 \theta = \sin^2 \theta \]

Thus, we can substitute these into our expression:
\[ \frac{\cos^2 \theta}{\sin^2 \theta} \]

3. **Recognize the cotangent function**:
The expression \( \frac{\cos^2 \theta}{\sin^2 \theta} \) can be rewritten as:
\[ \cot^2 \theta \]

4. **Substitute the value of cotangent**:
Since we are given that \( \cot \theta = \frac{7}{8} \), we can find \( \cot^2 \theta \):
\[ \cot^2 \theta = \left(\frac{7}{8}\right)^2 = \frac{49}{64} \]

5. **Final answer**:
Therefore, the value of the expression is:
\[ \frac{49}{64} \]

### Summary:
The value of \(\frac{(1 + \sin \theta)(1 - \sin \theta)}{(1 + \cos \theta)(1 - \cos \theta)}\) is \(\frac{49}{64}\).