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

Options

A. `(1)/(4)`
B. `(1)/(3)`
C. `(1)/(5)`
D. `(1)/(2)`

Answer

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 the sine function equals 1 at specific angles. The sine function equals 1 when its argument is \( \frac{\pi}{2} \) (or \( 90^\circ \)). Therefore, we can set up the equation:
\[ \sin^{-1}\left(\frac{1}{5}\right) + \cos^{-1}(x) = \frac{\pi}{2} \]

### Step 2: Isolate \( \cos^{-1}(x) \)
We can rearrange the equation to isolate \( \cos^{-1}(x) \):
\[ \cos^{-1}(x) = \frac{\pi}{2} - \sin^{-1}\left(\frac{1}{5}\right) \]

### Step 3: Use the Co-Function Identity
Using the co-function identity, we know that:
\[ \cos^{-1}(x) = \sin\left(\sin^{-1}\left(\frac{1}{5}\right)\right) \]
This means:
\[ x = \cos\left(\frac{\pi}{2} - \sin^{-1}\left(\frac{1}{5}\right)\right) = \sin\left(\sin^{-1}\left(\frac{1}{5}\right)\right) \]

### Step 4: Simplify the Expression
Since \( \sin\left(\sin^{-1}\left(\frac{1}{5}\right)\right) = \frac{1}{5} \), we find:
\[ x = \frac{1}{5} \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{\frac{1}{5}} \]