If `x=(1)/(5)`, the value of `cos(cos^(-1)x+2sin^(-1)x)` is :

Options

A. `-sqrt((24)/(25))`
B. `sqrt((24)/(25))`
C. `-(1)/(5)`
D. `(1)/(5)`

Answer

To solve the problem, we need to find the value of \( \cos(\cos^{-1}(x) + 2\sin^{-1}(x)) \) where \( x = \frac{1}{5} \).

### Step-by-step Solution:

1. **Substitute the value of \( x \)**:
\[ x = \frac{1}{5} \]
We need to evaluate:
\[ \cos(\cos^{-1}(\frac{1}{5}) + 2\sin^{-1}(\frac{1}{5})) \]

2. **Use the identity for \( \sin^{-1}(x) \) and \( \cos^{-1}(x) \)**:
We know that:
\[ \sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2} \]
Therefore, we can express \( \cos^{-1}(x) \) in terms of \( \sin^{-1}(x) \):
\[ \cos^{-1}(\frac{1}{5}) = \frac{\pi}{2} - \sin^{-1}(\frac{1}{5}) \]

3. **Substitute \( \cos^{-1}(\frac{1}{5}) \) in the expression**:
Now we can rewrite the expression:
\[ \cos\left(\left(\frac{\pi}{2} - \sin^{-1}(\frac{1}{5})\right) + 2\sin^{-1}(\frac{1}{5})\right) \]
Simplifying this gives:
\[ \cos\left(\frac{\pi}{2} + \sin^{-1}(\frac{1}{5})\right) \]

4. **Use the cosine of a sum identity**:
We know that:
\[ \cos\left(\frac{\pi}{2} + \theta\right) = -\sin(\theta) \]
Therefore:
\[ \cos\left(\frac{\pi}{2} + \sin^{-1}(\frac{1}{5})\right) = -\sin(\sin^{-1}(\frac{1}{5})) \]

5. **Evaluate \( -\sin(\sin^{-1}(\frac{1}{5})) \)**:
Since \( \sin(\sin^{-1}(x)) = x \), we have:
\[ -\sin(\sin^{-1}(\frac{1}{5})) = -\frac{1}{5} \]

6. **Final Result**:
Thus, the value of \( \cos(\cos^{-1}(\frac{1}{5}) + 2\sin^{-1}(\frac{1}{5})) \) is:
\[ -\frac{1}{5} \]

### Summary:
The final answer is:
\[ \cos(\cos^{-1}(\frac{1}{5}) + 2\sin^{-1}(\frac{1}{5})) = -\frac{1}{5} \]