Proof:
We know that to get catanary solutions, the solutions to our differential equation must of the form:
Our parameterization gives us the following:
Substitution yields:
Solving the above for theta gives solutions of the form:
This agrees with our results, since both answers are in the form of arctan. The reason that the arguments of the two inverse tangent functions don't agree, is that the argument of the hyperbolic cosine is actually much more complex, as it includes the initial conditions and other factors. Solving using this argument would yield a cleaner argument of the final inverse tangent function.
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