I want to know if this equation $(\cos x) ^{\frac{1}{\cos x}}+(\sin x) ^{\frac{1}{\sin x}}=1$ have integers solution , however wolfram alpha assumed all it solutions are real , but probably there is any method to disproof or proof that is has solutions in integers ?

  • $\begingroup$ So do you have a closed form way of expressing the two solutions in $[0,\pi/2]$? Let $x_1, x_2$ Be these two solutions. Then all the solutions to the equation are of the form $x_1+2n \pi$ and $x_2+2n \pi$ if I am not mistaken. $\endgroup$ – Mason May 27 '18 at 1:39
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    $\begingroup$ Did you graph it for $x \in (0, \frac 12 \pi)$? The function is undefined at $x=0$ but the limit as $x$ approaches $0$ from above is 1. There are two answers in the interval $x \in (0, \frac 12 \pi)$ and the chances are that you can get close to some large integer by adding the appropriate multiple of $2 \pi$ but never actually get an integer. $\endgroup$ – steven gregory May 27 '18 at 2:05
  • $\begingroup$ @stevengregory. Can you say your assumption explicitly? We are saying that one assumes that the solutions $x_1, x_2$ are transcendental and in fact one would guess that $x_i +2n\pi $ are never rational. What would be the way of expressing this algebraically? Something like $x_i \notin \mathbb{Q}[\pi]$? $\endgroup$ – Mason May 27 '18 at 2:21
  • $\begingroup$ Yes. But I don't know explicitly what the roots are, so I can't draw any actual conclusions. $\endgroup$ – steven gregory May 27 '18 at 4:48
  • $\begingroup$ My guess is that we cannot prove it, but I might be wrong. $\endgroup$ – Peter May 27 '18 at 9:00

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