1. Problem: Evaluate $\cos(625)$.\n2. Formula and rules: Cosine is periodic with period $2\pi$, so $$\cos(x+2\pi k)=\cos x\text{ for any integer }k$$.\n3. Reduce the angle modulo $2\pi$.\n4. Compute $k=\left\lfloor\dfrac{625}{2\pi}\right\rfloor=99$ and remainder $r=625-99\cdot 2\pi\approx 2.9646545892209$.\n5. Then $\cos(625)=\cos(r)$.\n6. Note $r=\pi-\delta$ with $\delta=\pi-r\approx 0.176938064368893$, so $\cos(625)=\cos(\pi-\delta)=-\cos(\delta)$.\n7. Use Taylor approximation $\cos(\delta)\approx 1-\dfrac{\delta^2}{2}+\dfrac{\delta^4}{24}$.\n8. Compute $\delta^2\approx 0.031307075$, $\delta^4\approx 0.000979$, so $\cos(\delta)\approx 1-0.015653537+0.00004079\approx 0.98438725$.\n9. Therefore $\cos(625)\approx -0.98438725$.\nFinal answer: $\cos(625)\approx -0.98438725$.\n