1. The problem is to evaluate the expression $$\cos^{-1}\left(\cos\left(\frac{7\pi}{8}\right)\right).$$ 2. Recall that the function $$\cos^{-1}(x)$$, also called arccosine, returns an angle in the range $$[0, \pi]$$. 3. The cosine function is periodic with period $$2\pi$$ and is symmetric about $$\pi$$, so $$\cos(\theta) = \cos(2\pi - \theta)$$. 4. Since $$\frac{7\pi}{8}$$ is in the second quadrant (between $$\frac{\pi}{2}$$ and $$\pi$$), and arccosine returns values in $$[0, \pi]$$, the value of $$\cos^{-1}\left(\cos\left(\frac{7\pi}{8}\right)\right)$$ is simply $$\frac{7\pi}{8}$$. 5. Therefore, the expression evaluates to $$\frac{7\pi}{8}$$. Final answer: $$\cos^{-1}\left(\cos\left(\frac{7\pi}{8}\right)\right) = \frac{7\pi}{8}.$$