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

Answer & Explanation

Johnny Cummings

Beginner2022-01-28Added 7 answers

It's a special case of the compound-angle formula $\mathrm{cos}(A+B)=\mathrm{cos}A\mathrm{cos}B-\mathrm{sin}A\mathrm{sin}B$. 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 $2{\mathrm{cos}}^{2}\frac{\theta}{2}-1$)