I was pondering about the different methods by which the half-angle identities for sine and cosine can be proved.
$$\left|\sin\left(\frac{x}2\right)\right|=\sqrt{\frac{1-\cos x}2}$$$$\left|\cos\left(\frac{x}2\right)\right|=\sqrt{\frac{1+\cos x}2}$$Besides the standard method (which uses the double angle formula for cosine), I have thought of $3$ others: the first being a geometrical method involving a circle, the second involving Euler's formula and the third involving vectors as elaborated below in my answer.
Are there any other creative ways to prove these identities?
Using the Double Angle Formula for Cosine$$\cos x=2\cos^2\left(\frac{x}2\right)-1=1-2\sin^2\left(\frac{x}2\right)$$$$\implies\cos^2\left(\frac{x}2\right)=\frac{1+\cos x}2, \sin^2\left(\frac{x}2\right)=\frac{1-\cos x}2$$$$\implies\left|\cos\left(\frac{x}2\right)\right|=\sqrt{\frac{1+\cos x}2}, \left|\sin\left(\frac{x}2\right)\right|=\sqrt{\frac{1-\cos x}2}$$