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Welcome to NRICH.

tanh²(x/2)=(cosh(x)-1)/(cosh(x)+1)

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By Anonymous on Friday, February 2, 2001 - 01:26 pm:

How can you prove the identity,
tanh²(x/2) == (cosh(x) - 1)/(cosh(x) + 1)

Thank you.

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By Kerwin Hui (Kwkh2) on Friday, February 2, 2001 - 02:06 pm:

One way to do this is by noting that

cosh x + 1 º ½(e^½x+e^-½x)²

and

cosh x - 1 º ½(e^½x-e^-½x)²

so substituting into RHS, you get LHS.

Kerwin

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By Anonymous on Friday, February 2, 2001 - 02:25 pm:

I don't fully understand how you got it?
where did the 1's go? How did cosh x + 1 become what you stated?

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By Kerwin Hui (Kwkh2) on Friday, February 2, 2001 - 02:35 pm:

OK, recall the definition of cosh x:

cosh x º ½(e^x+e^-x), for all x.

Hence, cosh x ± 1 º ½(e^x±2+e^-x)

Now the bracket on RHS is in the form a²±2ab+b², so result follows.

Kerwin

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By Anonymous on Friday, February 2, 2001 - 07:13 pm:

Thank you Kerwin

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