Is this proof of $\cos(z)=\cos^2\left(\frac z2\right)-\sin^2\left(\frac z2\right)$ correct?

Question

Is this proof of the following identity correct? $$\cos(z)=\cos^2\left(\frac z2\right)-\sin^2\left(\frac z2\right)$$

Answer 1

$$\left(\frac{e^{iz/2}+e^{-iz/2}}2\right)^2-\left(\frac{e^{iz/2}-e^{-iz/2}}{2i}\right)^2=\frac{e^{iz}+2+e^{-iz}+e^{iz}-2+e^{-iz}}4=\frac{e^{iz}+e^{-iz}}2.$$

or

$$2\left(\frac{e^{iz/2}+e^{-iz/2}}2\right)^2-1=2\frac{e^{iz}+2+e^{-iz}}4-1=\frac{e^{iz}+e^{-iz}}2.$$

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Alternatively,

$$(e^{iz/2})^2=\left(\cos\frac z2+i\sin\frac z2\right)^2=\cos^2\frac z2-\sin^2\frac z2+i\,2\cos\frac z2\sin\frac z2$$

gives you both

$$\cos z=\cos^2\frac z2-\sin^2\frac z2$$ and

$$\sin z=2\cos\frac z2\sin\frac z2.$$