Let $f(x)=(a x+b) \cos x+(c x+d) \sin x$ and $f'(x)=x \cos x$ for all $x$. Then,

$a=0, b=c=1, d=0$

We have,

$f(x)=(a x+b) \cos x+(c x+d) \sin x$

$\Rightarrow f'(x)=-(a x+b) \sin x+a \cos x+(c x+d) \cos x+c \sin x$

$\Rightarrow f'(x)=(c x+d+a) \cos x+(c-a x-b) \sin x$

But,  f'(x) = x cos x  for all x

∴  $(c x+d+a) \cos x+(c-a x-b) \sin x=x \cos x$  for all x

$\Rightarrow c=1, d+a=0, a=0$  and  $c-b=0$

$\Rightarrow a=0, b=c=1, d=0$