If `cos theta =4/5` then `sin^2 theta cos theta + cos^2 theta sin theta ` is equal to :

Options

A. `84/125`
B. `14/25`
C. `82/125`
D. `16/25`

Answer

To solve the problem, we need to find the value of the expression \( \sin^2 \theta \cos \theta + \cos^2 \theta \sin \theta \) given that \( \cos \theta = \frac{4}{5} \).

### Step-by-step Solution:

1. **Identify given values**:
We know that \( \cos \theta = \frac{4}{5} \).

2. **Find \( \sin \theta \)**:
We can use the Pythagorean identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting \( \cos^2 \theta \):
\[ \sin^2 \theta + \left(\frac{4}{5}\right)^2 = 1 \]
\[ \sin^2 \theta + \frac{16}{25} = 1 \]
\[ \sin^2 \theta = 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \]
Therefore, \( \sin \theta = \sqrt{\frac{9}{25}} = \frac{3}{5} \).

3. **Substitute values into the expression**:
We need to calculate:
\[ \sin^2 \theta \cos \theta + \cos^2 \theta \sin \theta \]
Substituting the values of \( \sin \theta \) and \( \cos \theta \):
\[ = \left(\frac{3}{5}\right)^2 \left(\frac{4}{5}\right) + \left(\frac{4}{5}\right)^2 \left(\frac{3}{5}\right) \]
\[ = \left(\frac{9}{25}\right) \left(\frac{4}{5}\right) + \left(\frac{16}{25}\right) \left(\frac{3}{5}\right) \]

4. **Calculate each term**:
The first term:
\[ \frac{9}{25} \cdot \frac{4}{5} = \frac{36}{125} \]
The second term:
\[ \frac{16}{25} \cdot \frac{3}{5} = \frac{48}{125} \]

5. **Combine the terms**:
\[ \frac{36}{125} + \frac{48}{125} = \frac{36 + 48}{125} = \frac{84}{125} \]

6. **Final answer**:
Therefore, the value of \( \sin^2 \theta \cos \theta + \cos^2 \theta \sin \theta \) is:
\[ \frac{84}{125} \]