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I need to show that: $\left(\frac{\cos\left(\frac{\pi}{2}\cdot\cos\theta\right)}{\sin\theta}\right)^2\simeq \sin^3(\theta)$

I used Matlab, their graphs are very similar. How to prove above estimate?

John Wayland Bales
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Son Tran Hoang
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  • The left hand side should be $\mid \sin(\theta)\mid^3$. – zoli May 30 '17 at 05:38
  • They are not even close to being almost equal. – Guy May 30 '17 at 05:42
  • From the graphs they appear to be approximately equal only in neighborhoods of multiples of $\frac{\pi}{2}$. But they can differ in between by as much as $0.05$. The approximation is better at odd multiples of $\frac{\pi}{2}$.https://www.desmos.com/calculator/ut816vneht – John Wayland Bales May 30 '17 at 05:44
  • The error is a third as big, around 0.017, for $|\sin(x)|^{2.63}$ – Empy2 May 30 '17 at 05:52
  • Well, it's not difficult to prove $c,\sin^2\theta\le\left(\frac{\cos\left(\frac{\pi}{2}\cdot\cos\theta\right)}{\sin\theta}\right)^2\le C,\sin^2\theta$ with some constants $0<c<C$, and that's certainly in contradiction with the claim, but as long as OP can't be bothered to react to comments, I can't be bothered to write that up. –  May 30 '17 at 18:57
  • @ProfessorVector Can you suggest me more? – Son Tran Hoang May 31 '17 at 08:52
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    First of all, you should make precise what the symbol $\simeq$ in your question is supposed to mean (a formal definition). That symbol is used for isomorphy, usually, but that doesn't make sense, here. –  May 31 '17 at 14:46

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