Solution
= Solution
The <total derivative> of $g$ sends a scalar increment $s$ to
$$
Dg(t)s=s\begin{pmatrix}\cos t\\-\sin t\end{pmatrix}.
$$
Since $\sin^2t+\cos^2t=1$, the <function composition> $f\circ g$ is constantly $1/2$, so $D(f\circ g)(t)=0$. Directly, the <Jacobian matrix> product is
$$
J_f(g(t))J_g(t)=
\begin{pmatrix}-\sin t/2&-\cos t/2\end{pmatrix}
\begin{pmatrix}\cos t\\-\sin t\end{pmatrix}=0.
$$
This verifies the <chain rule> at every $t$.