We have $sin(e^x-1) = sin (1+x+\frac{x^2}{2}+\frac{x^3}{6} + \frac{x^4}{24}$ ……………..-1)

$∴sin(e^x-1) = sin (x +\frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} …………)$

But $sinθ = θ–θ^3/3!+ θ^5/5! - …..$

$∴sin(e^x-1) = x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} +……. –1/6 (x+\frac{x^2}{2}+⋯…..)^3+……..$

$= x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} +……. –\frac{x^3}{6}–\frac{x^4}{24} + …….$

$= x + \frac{x^2}{2} - 5\frac{x^4}{24} + ……….$