ardeinduie

2022-01-27

where does the $\mathrm{cos}\left(\theta \right)=1-2{\mathrm{sin}}^{2}\left(\frac{\theta }{2}\right)$ come from?

Johnny Cummings

It's a special case of the compound-angle formula $\mathrm{cos}\left(A+B\right)=\mathrm{cos}A\mathrm{cos}B-\mathrm{sin}A\mathrm{sin}B$. Take $A=B=\frac{\theta }{2}$ so $\mathrm{cos}\theta ={\mathrm{cos}}^{2}\frac{\theta }{2}-{\mathrm{sin}}^{2}\frac{\theta }{2}$. This can be written in two equivalent forms using ${\mathrm{cos}}^{2}\frac{\theta }{2}+{\mathrm{sin}}^{2}\frac{\theta }{2}=1$, one being $1-2{\mathrm{sin}}^{2}\frac{\theta }{2}$ (the other is $2{\mathrm{cos}}^{2}\frac{\theta }{2}-1$)

eris0cg

$\mathrm{cos}\left(2\theta \right)=1-2{\mathrm{sin}}^{2}\left(\theta \right)$
Change $\theta \to \frac{\theta }{2}$

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