Find the value of
`cos( 2 cos^(-1) x + sin^(-1) x) " when " x = 1/5`

Options

A. ` (2 sqrt(6))/(5) `
B. `- (2 sqrt(6))/(5) `
C. ` (3 sqrt(6))/(5) `
D. None of these

Answer

To find the value of \( \cos(2 \cos^{-1} x + \sin^{-1} x) \) when \( x = \frac{1}{5} \), we can follow these steps:

### Step 1: Substitute the value of \( x \)
We start with the expression:
\[ \cos(2 \cos^{-1} x + \sin^{-1} x) \]
Substituting \( x = \frac{1}{5} \):
\[ \cos(2 \cos^{-1} \frac{1}{5} + \sin^{-1} \frac{1}{5}) \]

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

### Step 3: Substitute back into the expression
Now substituting this back into our expression:
\[ \cos(2 \cos^{-1} \frac{1}{5} + \left(\frac{\pi}{2} - \cos^{-1} \frac{1}{5}\right)) \]
This simplifies to:
\[ \cos\left(\cos^{-1} \frac{1}{5} + \frac{\pi}{2}\right) \]

### Step 4: Use the cosine addition formula
Using the cosine addition formula:
\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]
where \( a = \cos^{-1} \frac{1}{5} \) and \( b = \frac{\pi}{2} \):
\[ \cos\left(\cos^{-1} \frac{1}{5}\right) \cos\left(\frac{\pi}{2}\right) - \sin\left(\cos^{-1} \frac{1}{5}\right) \sin\left(\frac{\pi}{2}\right) \]
Since \( \cos\left(\frac{\pi}{2}\right) = 0 \) and \( \sin\left(\frac{\pi}{2}\right) = 1 \), we have:
\[ 0 - \sin\left(\cos^{-1} \frac{1}{5}\right) = -\sin\left(\cos^{-1} \frac{1}{5}\right) \]

### Step 5: Find \( \sin\left(\cos^{-1} \frac{1}{5}\right) \)
Using the identity \( \sin(\cos^{-1} x) = \sqrt{1 - x^2} \):
\[ \sin\left(\cos^{-1} \frac{1}{5}\right) = \sqrt{1 - \left(\frac{1}{5}\right)^2} = \sqrt{1 - \frac{1}{25}} = \sqrt{\frac{24}{25}} = \frac{\sqrt{24}}{5} \]
Thus:
\[ -\sin\left(\cos^{-1} \frac{1}{5}\right) = -\frac{\sqrt{24}}{5} \]

### Step 6: Simplify \( \sqrt{24} \)
We can simplify \( \sqrt{24} \):
\[ \sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6} \]
So:
\[ -\frac{\sqrt{24}}{5} = -\frac{2\sqrt{6}}{5} \]

### Final Answer
Therefore, the value of \( \cos(2 \cos^{-1} x + \sin^{-1} x) \) when \( x = \frac{1}{5} \) is:
\[ -\frac{2\sqrt{6}}{5} \]