If `sintheta=(7)/(25)`,then find the value of `sqrt((1-costheta)/(1+costheta))`.

Answer

To solve the problem where \( \sin \theta = \frac{7}{25} \) and we need to find the value of \( \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} \), we can follow these steps:

### Step 1: Use the Pythagorean Identity
We know from the Pythagorean identity that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Given \( \sin \theta = \frac{7}{25} \), we can find \( \cos^2 \theta \):
\[ \cos^2 \theta = 1 - \sin^2 \theta = 1 - \left(\frac{7}{25}\right)^2 \]

### Step 2: Calculate \( \sin^2 \theta \)
Calculating \( \sin^2 \theta \):
\[ \sin^2 \theta = \left(\frac{7}{25}\right)^2 = \frac{49}{625} \]

### Step 3: Substitute and Simplify
Now substitute \( \sin^2 \theta \) back into the equation for \( \cos^2 \theta \):
\[ \cos^2 \theta = 1 - \frac{49}{625} = \frac{625 - 49}{625} = \frac{576}{625} \]

### Step 4: Find \( \cos \theta \)
Taking the square root to find \( \cos \theta \):
\[ \cos \theta = \sqrt{\frac{576}{625}} = \frac{24}{25} \]

### Step 5: Substitute into the Expression
Now we substitute \( \cos \theta \) into the expression we need to evaluate:
\[ \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} = \sqrt{\frac{1 - \frac{24}{25}}{1 + \frac{24}{25}}} \]

### Step 6: Simplify the Expression
Calculating the numerator and denominator:
\[ 1 - \frac{24}{25} = \frac{1}{25}, \quad 1 + \frac{24}{25} = \frac{49}{25} \]
Thus, we have:
\[ \sqrt{\frac{\frac{1}{25}}{\frac{49}{25}}} = \sqrt{\frac{1}{49}} = \frac{1}{7} \]

### Final Answer
Therefore, the value of \( \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} \) is:
\[ \frac{1}{7} \]