If: `sin(sin^(-1)(1/5) + cos^(-1)x)=1`, then: x=

Options

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

Answer

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

### Step 1: Understand the equation
The equation states that the sine of the sum of two angles is equal to 1. We know that \( \sin(\theta) = 1 \) when \( \theta = \frac{\pi}{2} + 2n\pi \) for any integer \( n \). For simplicity, we will consider \( n = 0 \), so we have:
\[ \sin(\sin^{-1}(1/5) + \cos^{-1}(x)) = 1 \implies \sin^{-1}(1/5) + \cos^{-1}(x) = \frac{\pi}{2} \]

### Step 2: Rearranging 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 for inverse trigonometric functions
We know that:
\[ \cos^{-1}(x) + \sin^{-1}(x) = \frac{\pi}{2} \]
Thus, we can express \( \cos^{-1}(x) \) in terms of \( \sin^{-1}(x) \):
\[ \cos^{-1}(x) = \frac{\pi}{2} - \sin^{-1}(x) \]

### Step 4: Equate the two expressions for \( \cos^{-1}(x) \)
Now we can set the two expressions for \( \cos^{-1}(x) \) equal to each other:
\[ \frac{\pi}{2} - \sin^{-1}(1/5) = \frac{\pi}{2} - \sin^{-1}(x) \]

### Step 5: Simplify the equation
By simplifying, we can eliminate \( \frac{\pi}{2} \) from both sides:
\[ \sin^{-1}(1/5) = \sin^{-1}(x) \]

### Step 6: Solve for \( x \)
Since the inverse sine function is one-to-one in the range \([-1, 1]\), we can conclude:
\[ x = \frac{1}{5} \]

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