`sin(2tan^(-1) .(4)/(5))= ?`

Options

A. `(40)/(41)`
B. `(9)/(41)`
C. `(16)/(25)`
D. None of these

Answer

To solve the problem \( \sin(2 \tan^{-1}(\frac{4}{5})) \), we will use the formula for \( \sin(2 \tan^{-1}(x)) \):

\[ \sin(2 \tan^{-1}(x)) = \frac{2x}{1 + x^2} \]

### Step-by-step Solution:

1. **Identify \( x \)**:
Here, we have \( x = \frac{4}{5} \).

2. **Apply the formula**:
Substitute \( x \) into the formula:
\[ \sin(2 \tan^{-1}(\frac{4}{5})) = \frac{2 \cdot \frac{4}{5}}{1 + \left(\frac{4}{5}\right)^2} \]

3. **Calculate the numerator**:
\[ 2 \cdot \frac{4}{5} = \frac{8}{5} \]

4. **Calculate the denominator**:
First, compute \( \left(\frac{4}{5}\right)^2 \):
\[ \left(\frac{4}{5}\right)^2 = \frac{16}{25} \]
Now, add this to 1:
\[ 1 + \frac{16}{25} = \frac{25}{25} + \frac{16}{25} = \frac{41}{25} \]

5. **Combine the results**:
Now substitute the numerator and denominator back into the formula:
\[ \sin(2 \tan^{-1}(\frac{4}{5})) = \frac{\frac{8}{5}}{\frac{41}{25}} \]

6. **Simplify the fraction**:
Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \frac{8}{5} \cdot \frac{25}{41} = \frac{8 \cdot 25}{5 \cdot 41} = \frac{200}{205} = \frac{40}{41} \]

### Final Answer:
\[ \sin(2 \tan^{-1}(\frac{4}{5})) = \frac{40}{41} \]