In deriving the [half-angle formulas](http://www.sosmath.com/trig/douangl/douangl.html), my textbook first says: "Let's take the following identities:"

$$\cos^2\left(\frac a2\right)+\sin^2\left(\frac a2\right)=1;$$

$$\cos^2\left(\frac a2\right)-\sin^2\left(\frac a2\right)=\cos(a);$$

These identities I know. But then the texbook says "through addition and subtraction, we respectively arrive at:"

$$2\cos^2\left(\frac a2\right)=1+\cos(a)$$
$$2\sin^2\left(\frac a2\right)=1-\cos(a)$$

I failed to catch what exactly is added and what is substracted to arrive from the first two formulas to the second pair. Give me a hint, please.