1. **State the problem:** Simplify and understand the expression $$62.5 \cos(2\pi 60 t) \left( \frac{2\pi 60 e^{-t} \cos(2\pi 60 t) - e^{-t} \sin(2\pi 60 t)}{1 + (2\pi 60)^2} - \frac{2\pi 60}{1 + (2\pi 60)^2} \right)$$ 2. **Identify constants and terms:** - Angular frequency: $\omega = 2\pi 60$ - Denominator common term: $D = 1 + \omega^2$ 3. **Rewrite the expression using $\omega$ and $D$:** $$62.5 \cos(\omega t) \left( \frac{\omega e^{-t} \cos(\omega t) - e^{-t} \sin(\omega t)}{D} - \frac{\omega}{D} \right)$$ 4. **Distribute the denominator and combine terms inside the parentheses:** $$= 62.5 \cos(\omega t) \left( \frac{\omega e^{-t} \cos(\omega t) - e^{-t} \sin(\omega t) - \omega}{D} \right)$$ 5. **Factor out $e^{-t}$ from the numerator terms involving it:** $$= 62.5 \cos(\omega t) \frac{e^{-t} (\omega \cos(\omega t) - \sin(\omega t)) - \omega}{D}$$ 6. **Interpretation:** - The expression is a product of $62.5 \cos(\omega t)$ and a fraction with denominator $D$. - The numerator combines an exponentially decaying oscillatory term and a constant term $-\omega$. 7. **Summary:** The expression simplifies to $$\boxed{\frac{62.5 \cos(\omega t) \left(e^{-t} (\omega \cos(\omega t) - \sin(\omega t)) - \omega \right)}{1 + \omega^2}}$$ where $\omega = 2\pi 60$. This form is useful for analyzing the behavior over time $t$, showing oscillations modulated by exponential decay and constants.