$(20$ Points: $10$ Points each$)$ Find the Laplace Transform of $cos(t)$

a$.$ Using exponential form where

$cosx=\frac{e^{jx}+e^{-jx}}{2}$ and $L\{e^{at}\}=\frac{1}{s-a}$

b$.$ Using differentiation method where

$L\{f^{\text{'}}(t)\}=sL\{f(t)\}-f(0)$