Example 3 (Olympiad Training Problem from "Intermediate Mathematics" Issue 5, 2004) Let $a \in \mathbf{R}, A=\left\{x \mid 2^{1+x}+2^{1-x}=a\right\}, B=\{\sin \theta \mid$ $\theta \in \mathbf{R}\}$. If $A \cap B$ contains exactly one element, find the range of values for $a$.
Note that $B=[-1,1], 0 \in B$, obviously $a=2^{1+x}+2^{1-x} \geqslant 2 \sqrt{2^{1+x} \cdot 2^{1-x}}=4$, equality holds if and only if $2^{1+x}=2^{1-x}$, i.e., $x=0$. Therefore, $a=4$ satisfies the problem's requirements.
Next, we prove that $a>4$ do not satisfy the problem's requirements.
Assume $a>4$, and there is a non-zero solution $x_{1} \in B=[-1,1]$, then $a=2^{1+x_{1}}+2^{1-x_{1}}=2^{1+\left(-x_{1}\right)}+2^{1-\left(-x_{1}\right)}$, which means there is another non-zero solution $x_{2}=-x\left(x_{2} \neq x_{1}\right) \in B$, this contradicts the condition that $A \cap B$ has only one element. Hence, $a>4$ do not satisfy the problem's requirements.