Some of these are relatively trivial results of the sum and difference identities:

$\displaystyle \cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B)$

Put $\displaystyle A=B$:

$\displaystyle \cos(2A)=\cos^2(A)-\sin^2(A)$

then using Pythagoras's theorem in the form:

$\displaystyle \cos^2(x)+\sin^2(x)=1$

we get:

$\displaystyle \cos(2A)=2\cos^2(A)-1$

and

$\displaystyle \cos(2A)=1-2\sin^2(A)$

and by putting $\displaystyle a=2A$

$\displaystyle \cos(a)=2\cos^2(a/2)-1$

and

$\displaystyle \cos(a)=1-2\sin^2(a/2)$

RonL