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The principal value of $tan^{-1}(1)$ is the angle $\theta$ in the range $(-\frac{\pi}{2}, \frac{\pi}{2})$ such that $tan(\theta) = 1$. This occurs at $\theta = \frac{\pi}{4}$.
The principal value of $cos^{-1}(-\frac{1}{2})$ is the angle $\theta$ in the range $[0, \pi]$ such that $cos(\theta) = -\frac{1}{2}$. This occurs at $\theta = \frac{2\pi}{3}$.
The principal value of $sin^{-1}(-\frac{1}{\sqrt{2}})$ is the angle $\theta$ in the range $[-\frac{\pi}{2}, \frac{\pi}{2}]$ such that $sin(\theta) = -\frac{1}{\sqrt{2}}$. This occurs at $\theta = -\frac{\pi}{4}$.
Now, we add the principal values we found: $$tan^{-1}(1) + cos^{-1}(-\frac{1}{2}) + sin^{-1}(-\frac{1}{\sqrt{2}}) = \frac{\pi}{4} + \frac{2\pi}{3} - \frac{\pi}{4} = \frac{2\pi}{3}$$
Final Answer: $\frac{2\pi}{3}$
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