`cos[2cos^(-1).(1)/(5)+sin^(-1).(1)/(5)]`=

Options

A. `(2sqrt(6))/(5)`
B. `-(2sqrt(6))/(5)`
C. `(1)/(5)`
D. `-(1)/(5)`

Answer

To solve the expression \( \cos\left(2 \cos^{-1}\left(\frac{1}{5}\right) + \sin^{-1}\left(\frac{1}{5}\right)\right) \), we can follow these steps:

### Step 1: Use the Inverse Trigonometric Identity
We know that:
\[ \sin^{-1}(a) + \cos^{-1}(a) = \frac{\pi}{2} \]
For our case, we can express \( \cos^{-1}\left(\frac{1}{5}\right) \) in terms of \( \sin^{-1}\left(\frac{1}{5}\right) \):
\[ \cos^{-1}\left(\frac{1}{5}\right) = \frac{\pi}{2} - \sin^{-1}\left(\frac{1}{5}\right) \]

### Step 2: Substitute into the Cosine Expression
Now, substituting this into our expression:
\[ \cos\left(2 \cos^{-1}\left(\frac{1}{5}\right) + \sin^{-1}\left(\frac{1}{5}\right)\right) = \cos\left(2\left(\frac{\pi}{2} - \sin^{-1}\left(\frac{1}{5}\right)\right) + \sin^{-1}\left(\frac{1}{5}\right)\right) \]

### Step 3: Simplify the Argument
This simplifies to:
\[ \cos\left(\pi - 2 \sin^{-1}\left(\frac{1}{5}\right) + \sin^{-1}\left(\frac{1}{5}\right)\right) = \cos\left(\pi - \sin^{-1}\left(\frac{1}{5}\right)\right) \]

### Step 4: Use the Cosine Identity
Using the identity \( \cos(\pi - x) = -\cos(x) \):
\[ \cos\left(\pi - \sin^{-1}\left(\frac{1}{5}\right)\right) = -\cos\left(\sin^{-1}\left(\frac{1}{5}\right)\right) \]

### Step 5: Find Cosine using the Right Triangle
Let \( \theta = \sin^{-1}\left(\frac{1}{5}\right) \). In a right triangle, if \( \sin(\theta) = \frac{1}{5} \), then the opposite side is 1 and the hypotenuse is 5. The adjacent side can be found using the Pythagorean theorem:
\[ \text{Adjacent} = \sqrt{5^2 - 1^2} = \sqrt{25 - 1} = \sqrt{24} = 2\sqrt{6} \]
Thus,
\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{2\sqrt{6}}{5} \]

### Step 6: Substitute Back
Now substituting back into our expression:
\[ -\cos\left(\sin^{-1}\left(\frac{1}{5}\right)\right) = -\frac{2\sqrt{6}}{5} \]

### Final Answer
Thus, the final value is:
\[ \cos\left(2 \cos^{-1}\left(\frac{1}{5}\right) + \sin^{-1}\left(\frac{1}{5}\right)\right) = -\frac{2\sqrt{6}}{5} \]