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I know how to prove but i have difficult in choosing the signs.

sinh(y) = x = [exp(y)-exp(-y)]/2

there are two equations I can find

exp(2y) - 2x exp(y) -1 =0 OR exp(-2y) + 2xexp(-y) -1 =0

exp(y) = x +- sqrt(x^2 +1) & exp(-y) = -x +- sqrt(x^2 + 1)

if I select the signs of +

exp(y) = x + sqrt(x^2 +1) & exp(-y) = -x + sqrt(x^2 + 1)

then by substituting to cosh(y) = [exp(y) + exp(-y)] / 2

then, I can find the answer of

cosh(y)=sqrt(1+x^2)

But, what should I say to justify my actions of selecting the +ve sign.