5. If $M=\left\{z \left\lvert\, z=\frac{t}{1+t}+\mathrm{i} \frac{1+t}{t}\right., t \in \mathbf{R}, t \neq-1, t \neq 0\right\}$, $N=\{z|z=\sqrt{2} \cdot[\cos (\arcsin t)+\mathrm{i} \cos (\arccos t)], t \in \mathbf{R}| t \mid, \leqslant 1\}$, then the number of elements in $M \cap N$ is
A. 0
B. 1
C. 2
D. 4
$$
M=\{(x, y) \mid x y=1, x \neq 1\}, N=\left\{(x, y) \mid x^{2}+y^{2}=2, x \geqslant 0\right\} \text {, }
$$

From the function graphs, we can see that $M \cap N=\varnothing$. Therefore, the answer is $A$.