Step 1: Convert to a Sine:
Use $\cos\theta=\sin(\pi/2-\theta)$. For $\theta$ with $\pi/2-\theta$ in $[-\pi/2,\pi/2]$ the inverse sine just returns that angle, and otherwise it returns the equivalent angle in that range.
Step 2: Pairing:
The cosine values for $k$ and $k+4$ are negatives of each other ($\cos(\theta+\pi)=-\cos\theta$), and $\sin^{-1}$ is odd. So the terms for $k$ and $k+4$ cancel in pairs.
Step 3: Count:
Every block of 8 consecutive terms cancels in 4 pairs. After 2024 terms (253 blocks), only $k=2025\equiv1$ and $k=2026\equiv2$ remain: $\sin^{-1}(\cos\frac\pi4)+\sin^{-1}(\cos\frac\pi2)=\frac\pi4+0=\frac\pi4$. Option (C).
Final Answer:
Option (C).
\[ \boxed{\text{(C) } \frac{\pi}{4}} \]