Solve : `sin["sin"^(-1)1/5+cos^(-1)x]` = 1 .Find x.

Options

A. `(pi)/(2)`
B. 1
C. `-1`
D. None of these

Answer

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

### Step 1: Understand the equation
We know that \( \sin(\theta) = 1 \) when \( \theta = \frac{\pi}{2} + 2k\pi \) for any integer \( k \). In our case, we can set:
\[ \sin^{-1}(1/5) + \cos^{-1}(x) = \frac{\pi}{2} \]

### Step 2: Rearrange the equation
From the equation above, we can rearrange it to isolate \( \cos^{-1}(x) \):
\[ \cos^{-1}(x) = \frac{\pi}{2} - \sin^{-1}(1/5) \]

### Step 3: Use the identity
We know from trigonometric identities that:
\[ \cos^{-1}(x) + \sin^{-1}(x) = \frac{\pi}{2} \]
Thus, we can express \( \sin^{-1}(1/5) \) in terms of \( x \):
\[ \sin^{-1}(x) = \frac{\pi}{2} - \cos^{-1}(x) \]

### Step 4: Substitute the value
Now substituting \( \sin^{-1}(1/5) \) into the equation gives us:
\[ \sin^{-1}(1/5) = \frac{\pi}{2} - \cos^{-1}(x) \]

### Step 5: Solve for \( x \)
Using the identity \( \sin^{-1}(a) + \cos^{-1}(a) = \frac{\pi}{2} \), we can find:
\[ x = \cos(\cos^{-1}(x)) = \cos\left(\frac{\pi}{2} - \sin^{-1}(1/5)\right) \]
This simplifies to:
\[ x = \sin(\sin^{-1}(1/5)) = \frac{1}{5} \]

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