I've tried to solve the equation $\cot \theta = 2\cot 2\theta$ with the command 'Reduce' of Mathematica and obtained $\theta = n\pi$ as the solution with n an integer. But $\theta=n\pi$ is clearly a singularity in a cotangent function so this is puzzling.
I've realized that the above equation can be simplified to:
$\tan \theta=0$, and that is probably what Mathematica does to obtain the solution, but again, how $\theta=n\pi$ can be the solution if it is not in the domain of a $\cot$ function?
Is that an inconsistency in Mathematica or am I missing something?
Best Answer
If you expand $\cot2\theta$, you get $$ \cot\theta=\frac{\cot^2\theta-1}{\cot\theta} $$ that's clearly an inconsistent equation.
On the other hand, if you rewrite it as $$ \tan2\theta=2\tan\theta $$ and expand, you get $$ \frac{\tan\theta}{1-\tan^2\theta}=\tan\theta $$ that has the solutions $\tan\theta=0$, but they aren't solutions of the original equation, unless you allow $\infty$ as a value.
Depending on its implementation and conventions, a piece of software can give different results from “inconsistent”.