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## Article 21Circular Functions of Gudermanian.

The six hyperbolic functions of $u$ are expressible in terms of the six circular functions of its gudermanian; for since

 $\frac{{x}_{1}}{{a}_{1}}=coshu,\phantom{\rule{1em}{0ex}}\frac{{x}_{2}}{{a}_{2}}=coshv,$ (see Arts. 6, 7)

in which $u,v$ are the measures of corresponding hyperbolic and elliptic sectors, hence

 (46)

The gudermanian is sometimes useful in computation; for instance, if $sinhu$ be given, $v$ can be found from a table of natural tangents, and the other circular functions of $v$ will give the remaining hyperbolic functions of $u$. Other uses of this function are given in Arts. 22–26, 32–36.

Prob. 49.
Prove that
Prob. 50.
Prove
Prob. 51.
Prove
Prob. 52.
Show that $gdu$ and ${gd}^{-1}v$ are odd functions of $u,v$.
Prob. 53.
From the ﬁrst identity in 4, Prob. 17, derive the relation $tanh\frac{1}{2}u=tan\frac{1}{2}v$.
Prob. 54.
Prove and

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