If `tantheta=(8)/(15)` , the value of `(sqrt(1-sintheta))/(sqrt(1+sintheta))` is :

Options

A. `(1)/(5)`
B. `(2)/(5)`
C. `(3)/(5)`
D. 0

Answer

To solve the problem, we need to find the value of \(\frac{\sqrt{1 - \sin \theta}}{\sqrt{1 + \sin \theta}}\) given that \(\tan \theta = \frac{8}{15}\).

### Step-by-step Solution:

1. **Identify the sides of the triangle**:
Given \(\tan \theta = \frac{8}{15}\), we can represent this in a right triangle where the opposite side (perpendicular) is 8 and the adjacent side (base) is 15.

2. **Calculate the hypotenuse**:
Using the Pythagorean theorem:
\[ \text{Hypotenuse} = \sqrt{(\text{opposite})^2 + (\text{adjacent})^2} = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17 \]

3. **Find \(\sin \theta\)**:
\(\sin \theta\) is given by the ratio of the opposite side to the hypotenuse:
\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{8}{17} \]

4. **Substitute \(\sin \theta\) into the expression**:
Now substitute \(\sin \theta\) into the expression \(\frac{\sqrt{1 - \sin \theta}}{\sqrt{1 + \sin \theta}}\):
\[ \frac{\sqrt{1 - \frac{8}{17}}}{\sqrt{1 + \frac{8}{17}}} \]

5. **Simplify the numerator**:
Calculate \(1 - \frac{8}{17}\):
\[ 1 - \frac{8}{17} = \frac{17 - 8}{17} = \frac{9}{17} \]
Therefore, the numerator becomes:
\[ \sqrt{1 - \sin \theta} = \sqrt{\frac{9}{17}} = \frac{3}{\sqrt{17}} \]

6. **Simplify the denominator**:
Calculate \(1 + \frac{8}{17}\):
\[ 1 + \frac{8}{17} = \frac{17 + 8}{17} = \frac{25}{17} \]
Therefore, the denominator becomes:
\[ \sqrt{1 + \sin \theta} = \sqrt{\frac{25}{17}} = \frac{5}{\sqrt{17}} \]

7. **Combine the results**:
Now, substitute back into the expression:
\[ \frac{\frac{3}{\sqrt{17}}}{\frac{5}{\sqrt{17}}} = \frac{3}{5} \]

### Final Result:
The value of \(\frac{\sqrt{1 - \sin \theta}}{\sqrt{1 + \sin \theta}}\) is \(\frac{3}{5}\).

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