Answer
False
Work Step by Step
A transformation is said to be linear if $T(av+w)=aT(v)+T(w)$, for all $v,w$ in the given domain and for all scalars $a$.
This statement is false because
$\cos \frac{x}{2}\neq \frac{1}{2}\cos x$
Observe that
$\displaystyle D_x(\cos \frac{x}{2})=\frac{d(\cos \frac{x}{2})}{dx}=\frac{1}{2}\sin \frac{x}{2}$
and
$\displaystyle \frac{1}{2} D_x(\cos x)=\frac{1}{2}\sin x$
Since, $\frac{1}{2}\sin \frac{x}{2}\neq \frac{1}{2}\sin x$, the statement is false.
