Evaluate the exact numerical value of the given inverse trigonometric expression: $\sin\left[\cos^{-1} \cos\left(\frac{7\pi}{6}\right)\right]$.
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Always remember the principal value ranges:
- $\sin^{-1}(x) \in [-\frac{\pi}{2}, \frac{\pi}{2}]$
- $\cos^{-1}(x) \in [0, \pi]$
Never blindly cancel an inverse function with its regular function without verifying the angle range first!