Let f(x)=(ax+b)cosx+(cx+d)sinx and f|(x)=xcosx be an identity in x, then

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

$f^{|} (x) = a \cos x - (a x + b) \sin x + c . \sin x + (c x + d) . \cos x$

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

$but f^{|} (x) = x \cos x$

$c = 1 \Rightarrow - b + c = 0 \Rightarrow - b + 1 = 0 \Rightarrow b = 1$

$a = 0 \Rightarrow a + d = 0 \Rightarrow d = 0$

Let $f (x) = (a x + b) \cos x + (c x + d) \sin x$ and $f^{|} (x) = x \cos x$ be an identity in $x$ , then