Prove that sin ⁡ x + sin ⁡ y = 1 does not have integer solutionsSuppose x and y are angles measured in radians. Then how to show that the equation sin ⁡ x + sin ⁡ y = 1does not have a solution ( x , y ) ∈ N × N ?This question is prompted by curiosity. I don't have any ideas how it can be approached.

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Prove that   sin  ⁡  x  +  sin  ⁡  y  =  1 does not have integer solutionsSuppose x and y are angles measured in radians. Then how to show that the equation  sin  ⁡  x  +  sin  ⁡  y  =  1does not have a solution   (  x  ,  y  )  ∈      N    ×      N  ?This question is prompted by curiosity. I don&#039;t have any ideas how it can be approached.

Angelo Murray · Accepted Answer

No, and there is not even a solution for   (  x  ,  y  )  ∈      Q    ×      Q  .We can quickly exclude x=y, which would require that   sin  ⁡  x  =      1    2  , but that is only true for   x  =  n      π    6   for certain nonzero integers n, and none of these produce a rational. Similarly we can easily exclude x=0, y=0, or x=−y.Now, using Euler&#039;s formula, rewrite the equation to                    (*)                              e                      i            x                          +                  e                      i            y                          −                  e                      −            i            x                          −                  e                      −            i            y                          =        2        i        ⋅                  e          0                    and apply the Lindemann–Weierstrass theorem which in one formulation says that the exponentials of distinct algebraic numbers are linearly independent over the algebraic numbers. But   {  ±  i  x  ,  ±  i  y  ,  0  } are all algebraic and (by our assumptions so far) different, so (*) would be one of the linear relations that can&#039;t exist.This argument generalizes to show that the only algebraic number that can be written as a rational combination of sines of algebraic (radian) angles is 0.

Prove that sin ⁡ x + sin ⁡ y = 1 doe

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