Given x , y ∈ ( 0 , π 2 ], α &gt; 0 is a constant satisfying 0 &lt; x + y + α &lt; π,Also it is known that 1 sin ⁡ x + 1 sin ⁡ ( x + α ) = 1 sin ⁡ y + 1 sin ⁡ ( y + α ) How can we prove x=y "elegantly"?

Question

Given   x  ,  y  ∈  (  0  ,      π    2    ],   α  &amp;gt;  0 is a constant satisfying   0  &amp;lt;  x  +  y  +  α  &amp;lt;  π,Also it is known that      1          sin      ⁡      x        +      1          sin      ⁡      (      x      +      α      )        =      1          sin      ⁡      y        +      1          sin      ⁡      (      y      +      α      )      How can we prove x=y &quot;elegantly&quot;?

Sanaa Hinton · Accepted Answer

Put   x  =  x  +      α    2    −      α    2   to get  sin  ⁡  (  x  )  =  sin  ⁡  (  x  +      α    2    )  cos  ⁡  (      α    2    )  −  sin  ⁡  (      α    2    )  cos  ⁡  (  x  +      α    2    )and put   x  +  α  =  x  +      α    2    +      α    2   to get  sin  ⁡  (  x  +  α  )  =  sin  ⁡  (  x  +      α    2    )  cos  ⁡  (      α    2    )  +  sin  ⁡  (      α    2    )  cos  ⁡  (  x  +      α    2    )Now with   u  =  sin  ⁡  (  x  +      α    2    ) we have:      1          sin      ⁡      (      x      )        +      1          sin      ⁡      (      x      +      α      )        =  2  cos  ⁡  (      α    2    )      u                  u        2            −      (      sin      ⁡      (              α        2            )              )        2            Do the same with the other expression with   v  =  sin  ⁡  (  y  +      α    2    ). You get easily u=v or   u  v  =  −  (  sin  ⁡  (      α    2    )      )    2  , and it is easy, using   x  +  y  +  α  &amp;lt;  π, to finish

Shea Stuart · Accepted Answer

1          sin      ⁡      x        +      1          sin      ⁡      (      x      +      α      )        =      1          sin      ⁡      y        +      1          sin      ⁡      (      y      +      α      )      or,      1          sin      ⁡      x        −      1          sin      ⁡      y        =      1          sin      ⁡      (      y      +      α      )        −      1          sin      ⁡      (      x      +      α      )      or,            sin      ⁡      y      −      sin      ⁡      x              sin      ⁡      x      sin      ⁡      y        =            sin      ⁡      (      x      +      α      )      −      sin      ⁡      (      y      +      α      )              sin      ⁡      (      y      +      α      )      sin      ⁡      (      x      +      α      )      or,  −            2      cos      ⁡      (                        x          +          y                2            )      sin      ⁡      (                        x          −          y                2            )              sin      ⁡      x      sin      ⁡      y        =            2      cos      ⁡      (                        x          +          y                2            +      α      )      sin      ⁡      (                        x          −          y                2            )              sin      ⁡      (      y      +      α      )      sin      ⁡      (      x      +      α      )      or,  2  sin  ⁡  (            x      −      y        2    )      (                  cos        ⁡        (                              x            +            y                    2                )                    sin        ⁡        x        sin        ⁡        y              +                  cos        ⁡        (                              x            +            y                    2                +        α        )                    sin        ⁡        (        y        +        α        )        sin        ⁡        (        x        +        α        )              )    =  0It is very easy to say x=y from here. I hope you can do this last step.

Given x , y ∈ ( 0 , π 2 </mfrac> ] ,

Answered question

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