Step 1: Understanding the Concept:
We have an equation involving inverse trigonometric functions inside a sine function.
We can solve it by equating the angle to \( \sin^{-1}(1) \) and using standard inverse trigonometric identities.
Step 2: Key Formula or Approach:
1. \( \sin(\pi/2) = 1 \).
2. The identity \( \sin^{-1} \theta + \cos^{-1} \theta = \frac{\pi}{2} \) holds for all \( \theta \in [-1, 1] \).
Step 3: Detailed Explanation:
Given equation:
\[ \sin \left(\sin^{-1} \left(\frac{1}{5}\right) + \cos^{-1} x\right) = 1 \]
We know that the principal value of \( \sin^{-1}(1) \) is \( \frac{\pi}{2} \). Therefore, the angle inside the sine function must be \( \frac{\pi}{2} \) (or \( \frac{\pi}{2} + 2n\pi \), but working with principal values is sufficient here).
\[ \sin^{-1} \left(\frac{1}{5}\right) + \cos^{-1} x = \frac{\pi}{2} \]
Recall the fundamental inverse trigonometric identity:
\[ \sin^{-1} \theta + \cos^{-1} \theta = \frac{\pi}{2} \]
By comparing the two equations, we can clearly see that for the sum to be \( \frac{\pi}{2} \), the arguments of \( \sin^{-1} \) and \( \cos^{-1} \) must be identical.
Therefore,
\[ x = \frac{1}{5} \]
Step 4: Final Answer:
The value of x is \( \frac{1}{5} \).