Question

Help with this wild limit.

\[\lim_{\alpha \rightarrow 5\pi/2}\frac{ \cos(\csc (2\alpha)+\cot(2\alpha))-1 }{ \tan(\frac{ \cos^2\alpha }{ 1+\sin \alpha }) }\]

So for more clarity ,it's the limit as alpha goes to 5pi/2 of the following quotien

numerator: \[\cos(\csc(2\alpha)+\cot(2\alpha))-1\]
denominator: \[\tan(\frac{ \cos^2\alpha }{ 1+\sin \alpha })\]

Answer 1

yikes
i guess one first step would be to rewrite the denominator as
\[\tan(1-\sin(\alpha))\] maybe that might help

Answer 2

bet we can clean up the argument inside the numerator as well
i cheated and used the wolf, turns out tht
\[\csc(2\alpha)+\cot(2\alpha)=\cot(\alpha)\] although i don't see it right away, not familiar with "double angle" formulas for cosecant and cotangent

Answer 3

at least rewriting is as
\[\lim_{x\to \frac{5\pi}{4}}\frac{\cos(\cot(\alpha)-1}{\tan(1-\sin(\alpha))}\] will make l'hopital less ugly
i wonder if there is some gimmick to make this more obvious