Parametrization of $y^2 - x^2=1$ I have found parametrizations for the level curve $y^2-x^2=1$, however, I have a question regarding one of them.
From the Pythagorean trigonometric identity $\cos^2 x + \sin^2 x =1$ we obtain $$\sec^2 x - \tan^2 x =1, x \neq \pi(k+1/2), k \in \mathbb{Z},$$ $$\csc^2 x - \cot^2 x =1, x \neq k\pi, k \in \mathbb{Z}$$ by dividing by $\cos^2 x$ and $\sin^2 x$, respectively.
A parametrization for the given level curve is therefore $$\gamma(t) = (\tan t, \sec t), \ \ -\pi/2 < t < \pi/2, \quad \pi/2 < t < 3\pi/2.$$
A plot shows that this is indeed the case. However, for the other parametrization $$\gamma(t)=(\cot t, \csc t), \  0 < t < \pi, \quad \pi < t < 2\pi$$ it doesn't seem to work (plotting in Maple). Am I way off here on the second parametrization?
Other trigonometric functions satisfying this relationship is obviously $\cosh$ and $\sinh$ as in $\cosh^2 x - \sinh^2 x = 1, x \in \mathbb{R}$, but this only takes care of the upper branch of the level curve (by parametrizing $\gamma_1(t) = (\sinh t, \cosh t), t \in \mathbb{R}$). Would a complete parametrization then be a union of $\gamma_1(t)$ and $\gamma_2(t) = (\sinh t, -\cosh t), t \in \mathbb{R}$?
 A: I don't have Maple, but Mathematica does just fine with the first and second parametrizations:
ParametricPlot[{Tan[t], Sec[t]}, {t, -Pi/2, 3 Pi/2}, Exclusions -> Pi/2]

and
ParametricPlot[{Cot[t], Csc[t]}, {t, 0, 2 Pi}, Exclusions -> Pi]

give the same correct result.  The third parametrization using hyperbolic functions does only give the upper half for $\gamma_1$.  Unfortunately, trying to take the union of both halves no longer makes it a well-defined parametrization, since for each $t \in \mathbb R$, you now have two distinct ordered pairs $(x_1, y_1) \ne (x_2, y_2)$ on the curve.
As for why you are having problems in Maple, I can only state that your math is correct.  Whether your difficulty has to do with user error or a defect/shortcoming in Maple, I cannot say.
A: I've tried with Maxima:
plot2d([parametric,ctg(t), csc(t)],[t,0,%pi]);

gives a vertical line. Same even with smaller intervals such as:
plot2d([parametric,ctg(t), csc(t)],[t,0.05,3.09]);

Even with really small intervals like:
plot2d([parametric,ctg(t), csc(t)],[t,0.2,0.3]);

But
plot2d([parametric,1/tan(t), 1/sin(t)],[t,0.05,3.09]);

gives the upper branch.
My bet: some implementation-specific problem shared by Maple and Maxima.
A: Since $m:=\{y^2-x^2=1\mid x,y \text{ real numbers}\}$ is not connected there will be no continous curved $c$ defined on in interval $J$ with $c(J)=M$.
