1. **State the problem:** We are given the equation $$\frac{(1+x)e^t - (1-x)e^{-t}}{1+x^2} \cdot \frac{1-x^2}{1+x^2} = \varepsilon + x \varepsilon(x) t$$ and asked to analyze or simplify it. 2. **Rewrite the expression:** Focus on the left-hand side (LHS): $$\frac{(1+x)e^t - (1-x)e^{-t}}{1+x^2} \cdot \frac{1-x^2}{1+x^2}$$ 3. **Simplify the second fraction:** Note that $$1 - x^2 = (1-x)(1+x)$$ 4. **Combine the fractions:** The LHS becomes $$\frac{(1+x)e^t - (1-x)e^{-t}}{1+x^2} \times \frac{(1-x)(1+x)}{1+x^2} = \frac{((1+x)e^t - (1-x)e^{-t})(1-x)(1+x)}{(1+x^2)^2}$$ 5. **Simplify numerator:** Expand numerator terms: $$((1+x)(1-x)(1+x))e^t - ((1-x)(1-x)(1+x))e^{-t}$$ Note that $$(1+x)(1-x) = 1 - x^2$$ so numerator is $$ (1 - x^2)(1+x) e^t - (1 - x)^2 (1+x) e^{-t}$$ 6. **Factor out $(1+x)$:** $$ (1+x) \left[(1 - x^2) e^t - (1 - x)^2 e^{-t} \right]$$ 7. **Rewrite denominator:** Denominator is $$(1+x^2)^2$$ 8. **Final simplified form:** $$\frac{(1+x) \left[(1 - x^2) e^t - (1 - x)^2 e^{-t} \right]}{(1+x^2)^2} = \varepsilon + x \varepsilon(x) t$$ 9. **Interpretation:** The right side suggests a linear combination involving $\varepsilon$ and a function $\varepsilon(x)$ multiplied by $t$. This could represent a solution or approximation in terms of $x$ and $t$. **Answer:** The expression simplifies to $$\frac{(1+x) \left[(1 - x^2) e^t - (1 - x)^2 e^{-t} \right]}{(1+x^2)^2} = \varepsilon + x \varepsilon(x) t$$ which relates the given fractions and exponentials to the right-hand side involving $\varepsilon$ and $t$.