If sin `("sin"^(-1)(1)/(5)+ "cos"^(-1)x)=1`, then what is x equal to ?

Options

A. 0
B. 1
C. `(4)/(5)`
D. `(1)/(5)`

Explanation

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

### Step 1: Understand the equation
We know that \( \sin(y) = 1 \) when \( y = \frac{\pi}{2} + 2k\pi \) for any integer \( k \). In this case, we can focus on the principal value, so we set:
\[ \sin^{-1}\left(\frac{1}{5}\right) + \cos^{-1}(x) = \frac{\pi}{2} \]

### Step 2: Rearranging the equation
From the equation above, we can isolate \( \cos^{-1}(x) \):
\[ \cos^{-1}(x) = \frac{\pi}{2} - \sin^{-1}\left(\frac{1}{5}\right) \]

### Step 3: Use the identity for cosine
We know that:
\[ \cos^{-1}(x) = \sin^{-1}(y) \implies x = \sqrt{1 - y^2} \]
where \( y = \sin^{-1}\left(\frac{1}{5}\right) \).

### Step 4: Find \( \sin\left(\sin^{-1}\left(\frac{1}{5}\right)\right) \)
We have:
\[ \sin\left(\sin^{-1}\left(\frac{1}{5}\right)\right) = \frac{1}{5} \]

### Step 5: Use the Pythagorean identity
Using the identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \):
\[ \cos^2\left(\sin^{-1}\left(\frac{1}{5}\right)\right) = 1 - \left(\frac{1}{5}\right)^2 = 1 - \frac{1}{25} = \frac{24}{25} \]
Thus,
\[ \cos\left(\sin^{-1}\left(\frac{1}{5}\right)\right) = \sqrt{\frac{24}{25}} = \frac{\sqrt{24}}{5} = \frac{2\sqrt{6}}{5} \]

### Step 6: Relate \( x \) to the cosine value
Since we have:
\[ x = \cos\left(\cos^{-1}(x)\right) = \sin\left(\sin^{-1}\left(\frac{1}{5}\right)\right) = \frac{2\sqrt{6}}{5} \]

### Step 7: Final answer
Thus, the value of \( x \) is:
\[ x = \frac{2\sqrt{6}}{5} \]