Difference between revisions of "Relationship between coth, inverse Gudermannian, and csc"
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| − | + | ==Theorem== | |
| − | + | The following formula holds: | |
$$\mathrm{coth}(\mathrm{gd}^{-1}(x))=\csc(x),$$ | $$\mathrm{coth}(\mathrm{gd}^{-1}(x))=\csc(x),$$ | ||
where $\mathrm{coth}$ is the [[coth|hyperbolic cotangent]], $\mathrm{gd}^{-1}$ is the [[inverse Gudermannian]], and $\csc$ is the [[cosecant]]. | where $\mathrm{coth}$ is the [[coth|hyperbolic cotangent]], $\mathrm{gd}^{-1}$ is the [[inverse Gudermannian]], and $\csc$ is the [[cosecant]]. | ||
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| − | + | ==Proof== | |
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| − | + | ==References== | |
Revision as of 04:37, 6 June 2016
Theorem
The following formula holds: $$\mathrm{coth}(\mathrm{gd}^{-1}(x))=\csc(x),$$ where $\mathrm{coth}$ is the hyperbolic cotangent, $\mathrm{gd}^{-1}$ is the inverse Gudermannian, and $\csc$ is the cosecant.
