The Maclaurin's series expansion of e^{sin x} is

f(x) = e^{sin x} ⇒ f(0) = e⁰ = 1

f'(x) = esin x (cos x) ⇒ f'(0) = 1

f''(x) = esin x cos²x + esin x (-sin x)

f''(0) = 1

= esin x [cos²x - sin x]

f'''(x) = esin x [-sin 2x - cos x] + esin x  [cos³x - sin x cos x]

f'''(x) = -1 + 1 = 0

f^iv(x) = esin x  [-2 cos 2x + sin x] + esin x  cos x [-sin 2x - cos x]

\(+ e^{\sin x } \left[ 3 \cos^2 x (-\sin x) - \frac{\cos 2x}{2} \times 2 \right]\)

+ esin x [cos⁴ x - sin x cos² x]

f^iv(0) = -2 -1 -1 + 1 = -3

Substitue in Maclaurin Series

\(e^{\sin x} = 1 + x + \frac{x^2}{2 } + \frac{x^3}{3!} (0) + \frac{x^4}{4!} \times (-3) + ....\)

\(= 1+ x + \frac{x^2}{2} - \frac{x^4}{8} + ....\)