# Does this:$(\cos x) ^{\frac{1}{\cos x}}+(\sin x) ^{\frac{1}{\sin x}}=1$ have integers solution?

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 ?

• 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. – Mason May 27 '18 at 1:39
• 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. – steven gregory May 27 '18 at 2:05
• @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 $x_i \notin \mathbb{Q}[\pi]$? – Mason May 27 '18 at 2:21
• Yes. But I don't know explicitly what the roots are, so I can't draw any actual conclusions. – steven gregory May 27 '18 at 4:48
• My guess is that we cannot prove it, but I might be wrong. – Peter May 27 '18 at 9:00