Given, $(\cos 4+i\sin 4+1{)}^{2020}$

Now using, $1+\cos 2A=2{\cos }^{2}A$ and $\sin 2A=2\sin A\cos A$, we get

${\left(2{\cos }^{2}2+2i\sin 2\cos 2\right)}^{2020}$

$\Rightarrow {\left[2\cos 2\left(\cos 2+i\sin 2\right)\right]}^{2020}$

Now, By using De Moivre's Theorem, we get

$\Rightarrow 2^{2020}{\cos }^{2020}2\left[\cos \left(2\times 2020\right)+i\sin \left(2\times 2020\right)\right]$

$\Rightarrow 2^{2020}{\cos }^{2020}2\left[\cos \left(4040\right)+i\sin \left(4040\right)\right]$

Hence, the Real part is $2^{2020}{\cos }^{2020}2\cos 4040$