f(x) = cos x

f(0) = 1

First dervative:

f '(x) = −sin x

f '(0) = −sin 0 = 0

Second dervative:

f ''(x) = −cos x

f ''(0) = −cos 0 = -1

Third dervative:

f '''(x) = sin x

f '''(0) = sin 0 = 0

Fourth dervative:

f iv(x) = cos x

f iv(0) = cos 0 = 1

When substituting these values into the expansion formula, we obtain:

`cos x=1-1/2x^2` `+1/24x^4` `-1/720x^6` `+1/40320x^8-...`