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I have a problem with the following equation that is related to Cerenkov counter. You can find it at page 56 of "Introduction of High Energy Physics" by Perkins.

The equation is the following:

[tex]sin ^{2} (\theta _1)=1-\frac{1}{\beta ^{2}_{1}n^2}\approx\frac{m^{2}_{2}-m^{2}_1}{p^2}[/tex]

I do know why : [tex]sin ^{2}(\theta _1})= 1-\frac{1}{\beta ^{2}_{1}n^{2}}[/tex]

since it comes out from [tex]cos(\theta _{1})=\frac{1}{\beta n}[/tex]

but I don't know where : [tex]1-\frac{1}{\beta ^{2}_{1}n^2}\approx \frac{m^{2}_{2}-m^{2}_1}{p^2}[/tex]

comes from...

Could anyone explain this to me ?

Thank you very much in advance for any help !

Erwin