The formula for the hyperbolic cotangent is derived from the definitions of hyperbolic functions.
The hyperbolic cotangent, denoted as coth(x), is defined as the ratio of the hyperbolic cosine and hyperbolic sine functions:
coth(x)=sinh(x)cosh(x)The hyperbolic cosine and sine functions are defined as follows:
cosh(x)=2ex+e−x sinh(x)=2ex−e−xBy substituting these definitions into the formula for coth(x), we get:
coth(x)=2ex−e−x2ex+e−x=ex−e−xex+e−xTo simplify this expression, we can multiply both the numerator and the denominator by ex:
coth(x)=e2x−1e2x+1This provides us with the formula for the hyperbolic cotangent. Additionally, it can also be expressed in terms of exponential functions as follows:
coth(x)=ex−e−xex+e−x=e2x−1e2x+1=e−2x−11+e−2xThese various forms highlight the relationships between the hyperbolic cotangent and exponential functions.
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