We know that the principal value branch of $\sec^{-1}(x)$ is $[0, \pi] - \{\frac{\pi}{2}\}$. Since $\sec(\frac{\pi}{4}) = \sqrt{2}$, we have: $$\sec^{-1}(\sqrt{2}) = \frac{\pi}{4}$$
We use the property $\text{cosec}^{-1}(-x) = -\text{cosec}^{-1}(x)$. Thus: $$\text{cosec}^{-1}(-\sqrt{2}) = -\text{cosec}^{-1}(\sqrt{2})$$ Since $\text{cosec}(\frac{\pi}{4}) = \sqrt{2}$, we have $\text{cosec}^{-1}(\sqrt{2}) = \frac{\pi}{4}$. Therefore: $$\text{cosec}^{-1}(-\sqrt{2}) = -\frac{\pi}{4}$$
Substitute the values back into the original expression: $$\frac{\pi}{4} + 2\left(-\frac{\pi}{4}\right) = \frac{\pi}{4} - \frac{2\pi}{4} = -\frac{\pi}{4}$$
Final Answer: -\frac{\pi}{4}
AI generated content. Review strictly for academic accuracy.