1. Theoretical Principle: The identity states that for any value of $x$ within the closed interval $[-1, 1]$, the sum of the arcsine and arccosine of that value always equals a constant right angle.
2. Mathematical Proof: Let $\sin^{-1} x = \theta$. By the definition of inverse functions, this implies:
$$x = \sin \theta$$
Using the co-function identity $\sin \theta = \cos(\frac{\pi}{2} - \theta)$, we can write:
$$x = \cos\left(\frac{\pi}{2} - \theta\right)$$
Applying the inverse cosine function to both sides:
$$\cos^{-1} x = \frac{\pi}{2} - \theta$$
3. Final Summation: Now, substitute $\theta = \sin^{-1} x$ back into the equation:
$$\cos^{-1} x = \frac{\pi}{2} - \sin^{-1} x$$
$$\cos^{-1} x + \sin^{-1} x = \frac{\pi}{2}$$
This holds true for all $x \in [-1, 1]$.