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: