If for real values of $x,$ $\cos \theta =x+\frac{1}{x}$ then

$x+\frac{1}{x}\ge 2 for x>0$ $or x+\frac{1}{x}\le -2 if x<0$ 

$$\begin{gathered} \Rightarrow \cos \theta \ge 2 or \cos \theta \le -2 \\ This is not possible \sin ce -1\le \cos \theta \le 1 \\ \Rightarrow no value of \theta is possible \end{gathered}$$