Statement 1: $$\cos 2x = \cos^2 x - \sin^2 x$$
Proof 1: Use the Angle Addition Formula for Cosine:
$$\cos(2x) = \cos(x + x) = \cos(x)\cos(x) - \sin(x)\sin(x) = \cos^2(x) - \sin^2(x)$$
Statement 2: $$\cos 2x = 1 - 2\sin^2 x$$
Proof 2: We can prove that
$$\cos^2(x) - \sin^2(x) = 1 - 2\sin^2(x)$$
because the left-hand side is equivalent to $$\cos(2x)$$. Add $$2\sin^2(x)$$ to both sides of the equation:
$$\cos^2(x) + \sin^2(x) = 1$$
This is obviously true.
Statement 3: $$\cos 2x = 2\cos^2 x - 1$$
Proof: It suffices to prove that
$$1 - 2\sin^2 x = 2\cos^2 x - 1$$
Add $$1$$ to both sides of the equation:
$$2 - 2\sin^2 x = 2\cos^2 x$$
Now add $$2\sin^2 x$$ to both sides of the equation:
$$2 = 2\cos^2 x + 2\sin^2 x = 2(\cos^2 x + \sin^2 x) = 2$$
Therefore this equality also holds.