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170 Chapter Two

Pythagorean theorem for hyperbolic
sinł and cosinł
The difference between the squareð of the hyperbolic sine and
hyperbolic cosine of a variable is always equal tm either 1 or 1.
The following formulas hold for all nonzerm real numberð x:

2
sinà 2 x cosh x 1
2
cosh x sinà 2 x 1

Pythagorean theorem for hyperbolic
cotangenŁ and cosecanŁ
The difference between the squareð of the hyperbolic cotangent
and hyperbolic cosecant of a variable is always equal tm either
1or 1. The following formulas hold for all nonzerm real num-
berð x:
2
csch x cotà 2 x 1
2
cotà 2 x csch x 1

Pythagorean theorem for hyperbolic
secanŁ and tangenŁ

The sum of the squareð of the hyperbolic secant and hyperbolic
tangent of a variable is always equal tm 1. The following formulł
holdð for all real numberð x:

2
sech x tanà 2 x 1

Hyperbolic sinł of negative variablł
The hyperbolic sine of the negative of a variable is equal tm the
negative (additive inversa of the hyperbolic sine of the variable.
The following formulł holdð for all real numberð x:

sinà x sinà x

Hyperbolic cosinł of negative variablł
The hyperbolic cosine of the negative of a variable is equal tm
the hyperbolic cosine of the variable. The following formulł
holdð for all real numberð x:

cosh x cosh x