The set of all values of a for which the function $f(x)=\left(a^2-3 a+2\right)\left(\cos ^2 \frac{x}{4}-\sin ^2 \frac{x}{4}\right)+(a-1) x+\sin 1$ does not possess critical points, is 

$(0,1) \cup(1,4)$

We have,

$f(x)=\left(a^2-3 a+2\right)\left\{\cos ^2 \frac{x}{4}-\sin ^2 \frac{x}{4}\right\}+(a-1) x+\sin 1 $

$\Rightarrow f(x)=(a-1)(a-2) \cos \frac{x}{2}+(a-1) x+\sin 1$

$\Rightarrow f^{\prime}(x)=\frac{-1}{2}(a-1)(a-2) \sin \frac{x}{2}+(a-1)$

$\Rightarrow f^{\prime}(x)=(a-1)\left\{1-\frac{(a-2)}{2} \sin \frac{x}{2}\right\}$

If $f(x)$ does not possess critical points, then

$f^{\prime}(x) \neq 0$ for any $x \in R$

$\Rightarrow (a-1)\left\{1-\frac{(a-2)}{2} \sin \frac{x}{2}\right\} \neq 0$ for any $x \in R$

$\Rightarrow a \neq 1$ and $1-\left(\frac{a-2}{2}\right) \sin \frac{x}{2}=0$ must not have any solution in R

$\Rightarrow a \neq 1$ and $\sin \frac{x}{2}=\frac{2}{a-2}$ is not solvable in R

$\Rightarrow a \neq 1$ and $\left|\frac{2}{a-2}\right|>1$

$\Rightarrow a \neq 1$ and $|a-2|<2$

$\Rightarrow a \neq 1$ and $-2<a-2<2$

$\Rightarrow a \neq 1$ and $0<a<4$

$\Rightarrow a \in(0,1) \cup(1,4)$.