If for real values of x,

Given for real value of $x, \cos \theta=x+\frac{1}{x}$

i. e $\cos \theta=\frac{x^{2}+1}{x}$

$\Rightarrow x \cos \theta=x^{2}+1$

$\Rightarrow x^{2}-x \cos \theta+1=0$

for $x \in \mathbb{R}$ i. e roots should be real

i. e $b^{2}-4 a c \geq 0$

i. e $\cos ^{2} \theta-4(1)(1) \geq 0$

i. e $\cos ^{2} \theta \geq 4$

which is not possible $\quad(\because-1 \leq \cos \theta \leq 1)$

∴ No such value of θ is possible

Hence, the correct answer is option D.