If `costheta=(3)/(5)` , then the value of `sintheta.sectheta.tantheta` is

Options

A. `(9)/(16)`
B. `(16)/(9)`
C. `(3)/(4)`
D. `(4)/(3)`

Answer

To solve the problem, we need to find the value of \( \sin \theta \cdot \sec \theta \cdot \tan \theta \) given that \( \cos \theta = \frac{3}{5} \).

### Step-by-Step Solution:

1. **Identify the relationship between trigonometric functions**:
We know that:
\[ \sec \theta = \frac{1}{\cos \theta} \]
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Therefore, we can express \( \sin \theta \cdot \sec \theta \cdot \tan \theta \) as:
\[ \sin \theta \cdot \sec \theta \cdot \tan \theta = \sin \theta \cdot \frac{1}{\cos \theta} \cdot \frac{\sin \theta}{\cos \theta} \]
This simplifies to:
\[ = \frac{\sin^2 \theta}{\cos^2 \theta} \]

2. **Use the Pythagorean identity**:
We know from the Pythagorean identity that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Given \( \cos \theta = \frac{3}{5} \), we can find \( \sin^2 \theta \):
\[ \sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25} \]

3. **Calculate \( \sin^2 \theta \) and \( \cos^2 \theta \)**:
We already have:
\[ \sin^2 \theta = \frac{16}{25} \]
Now calculate \( \cos^2 \theta \):
\[ \cos^2 \theta = \left(\frac{3}{5}\right)^2 = \frac{9}{25} \]

4. **Substitute back into the expression**:
Now substitute \( \sin^2 \theta \) and \( \cos^2 \theta \) into the expression:
\[ \frac{\sin^2 \theta}{\cos^2 \theta} = \frac{\frac{16}{25}}{\frac{9}{25}} = \frac{16}{9} \]

5. **Final answer**:
Thus, the value of \( \sin \theta \cdot \sec \theta \cdot \tan \theta \) is:
\[ \frac{16}{9} \]