 Solution

For every integer $j\ge 0$, the $j$th derivative with respect to $t$ of $\cos (t\cos x)$ is $(\cos x{)}^j$ times a sine or cosine with a sign. Its absolute value is at most $1$ on the finite interval $[0,\pi /2]$. This is an integrable dominating function independent of $t$. Repeated differentiation under the integral sign is therefore justified for every order, giving $J_0\in C^{\infty }(\mathbb{R})$.