(Solved): 7. Use complex exponentials to show the trigonometric identities: a. cos(+)=cos()cos()si ...

7. Use complex exponentials to show the trigonometric identities: a. $$\cos (?+?)=\cos (?)\cos (?)?\sin (?)\sin (?)$$ b. $$\cos (???)=\cos (?)\cos (?)+\sin (?)\sin (?)$$ Hint: write the left hand side of each equation as the real part of a complex exponential. 8. Impulse functions. Simplify the following equations to remove any differentiators and integrals: a. $$\frac{du(t?a)}{dt}=$$ b. $$u(t)(?(t+1)??(t?1))$$ c. $${?}_{??}^{?}{e}^{?\mathrm{san}{?}_0}?(t)dt=$$ d. Define clearly what the difference is between the following results: $${?}_{?}^{?1}f(t)?(t?a)dt\text{ and }f(t)?(t?a)$$