How come that $(\cos2x + 1) = (2 \cos^2 x – 1 + 1)?$ I really am not able to find any identity which results in this expression, any help regarding how we obtain the right-hand side of the equation  $$\cos2x + 1 = 2 \cos^2 x$$ would be really appreciated. :)

Using the identities $$\cos{(a+b)}=\cos{(a)} \cos{(b)}-\sin{(a)} \sin{(b)}$$ and $$\sin^2{(x)}+\cos^2{(x)}=1 \Rightarrow \sin^2{(x)}=1-\cos^2{(x)}$$ we have the following:
$$\cos{(2x)}+1=\cos{(x+x)}+1=\cos{(x)} \cos{(x)}-\sin{(x)} \sin{(x)}+1= \\ \cos^2{(x)}-\sin^2{(x)}+1=\cos^2{(x)}-(1-\cos^2{(x)})+1=2\cos^2{(x)}-1+1=2 \cos^2{(x)}$$
