Example 10 Solve the equation $\cos \frac{x^{2}+x}{2}-2^{x-1}-2^{-x-1}=0$.

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Example 10 Solve the equation $\cos \frac{x^{2}+x}{2}-2^{x-1}-2^{-x-1}=0$.
Analysis: Transform the original equation into
$$\cos \frac{x^{2}+x}{2}=2^{x-1}+2^{-x-1} .$$

It seems difficult to proceed with further transformations, so we might consider "inequality" instead of "equality". Thus, we have
$$2^{x-1}+2^{-x-1} \geqslant 2 \sqrt{2^{x-1} \cdot 2^{-x-1}}=1$$

and $\cos \frac{x^{2}+x}{2} \leqslant 1$.
Therefore, for the original equation to hold, the inequalities (1) and (2) must both be equalities, which occurs if and only if $x-1=-x-1$, i.e., $x=0$.

Hence, the original equation has a unique real solution $x=0$.