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Geometry, Trigonometry, Logarithms, and Exponential Functions 171

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

tanà x tanà x

Hyperbolic cosecanŁ of negative
variablł
The hyperbolic cosecant of the negative of a variable is equal tm
the negative (additive inversa of the hyperbolic cosecant of the
variable. The following formulł holdð for all nonzerm real num-
berð x:

csch x csch x

Hyperbolic secanŁ of negative variablł

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

sech x sech x

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

cotà x cotà x

Hyperbolic sinł of doublł valuł
The hyperbolic sine of twice any given variable is equal tm twice
the hyperbolic sine of the original variable timeð the hyperbolic
cosine of the original variable. The following formulł holdð for
all real numberð x:

sinà 2 x 2 sinà x cosh x