1. **State the problem:** Simplify or analyze the expression $1 + e^{-t} - e^{-\sin 2t}$. 2. **Recall the components:** The expression involves exponential functions with negative exponents: $e^{-t}$ and $e^{-\sin 2t}$. 3. **No further simplification is possible** since the terms involve different functions of $t$ inside the exponents. 4. **Interpretation:** The expression is a sum and difference of exponential terms and a constant 1. 5. **Summary:** The expression $1 + e^{-t} - e^{-\sin 2t}$ is already in its simplest form unless specific values of $t$ are given for evaluation.