`sinh^2(x) = (-1+cosh(2x))/2`
proof:
LHS=>
`sinh^2(x) =(( e^x - e^(-x))/2)^2 `
= `((e^x - e^(-x))^2)/4`
= `((e^x)^2 - 2e^x e^(-x) + (e^-x)^2)/4`
= `(e^(2x) + e^(-2x )- 2)/4`
=` (-2 + e^(2x) + e^(-2x))/4 `
= `((-2 + e^(2x) + e^(-2x))/2)/2`
= `(-1 + (e^(2x) + e^(-2x))/2)/2 `
= `(-1 + cosh 2x)/2 `
= RHS
as LHS=RHS
so,
`sinh^2(x) = (-1+cosh(2x))/2`
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