Solution
= Solution
Write the <sine> argument as $\pi/2+\delta$, with $\delta=\tfrac\pi2(e^x-1)=\tfrac\pi2x+\tfrac\pi4x^2+O(x^3)$. The <sine addition formula> gives $\sin(\pi/2+\delta)=\cos\delta$. Its <Maclaurin series> is $1-\delta^2/2+O(\delta^4)$, so
$$
\boxed{\sin\left(\frac{\pi e^x}{2}\right)=1-\frac{\pi^2}{8}x^2+O(x^3).}
$$
There is no linear term because the outer <sine> has zero <derivative> at $\pi/2$.