To be honest,  I am not really sure what this question means ..but
  
 \(\frac{e^{ix}}{e^{iy}}\\ =\frac{cos(x)+isin(x)}{cos(y)+isin(y)}\\ =\frac{cos(x)+isin(x)}{cos(y)+isin(y)}*\frac{cos(y)-isin(y)}{cos(y)-isin(y)}\\~\\ =\frac{cos(x)cos(y)-icos(x)sin(y)+isin(x)cos(y) +sin(x)sin(y) }{cos^2(y)+sin^2(y)}\\~\\ =\frac{cos(x)cos(y) +sin(x)sin(y) +i[-cos(x)sin(y)+sin(x)cos(y) ] }{1}\\~\\ =cos(x-y)+isin(x-y)\)
  
 ALSO
 \(\frac{e^{ix}}{e^{iy}}\\ =e^{ (ix-iy) }\\ =e^{ i(x-y) }\\ =cos(x-y)+isin(x-y)\)
  
 I must be almost finished but i don't get what the question actually wants.   
  
  
  
  
 LaTex
 \frac{e^{ix}}{e^{iy}}\\
 =\frac{cos(x)+isin(x)}{cos(y)+isin(y)}\\
 =\frac{cos(x)+isin(x)}{cos(y)+isin(y)}*\frac{cos(y)-isin(y)}{cos(y)-isin(y)}\\~\\
 =\frac{cos(x)cos(y)-icos(x)sin(y)+isin(x)cos(y)    +sin(x)sin(y) }{cos^2(y)+sin^2(y)}\\~\\
 =\frac{cos(x)cos(y)  +sin(x)sin(y)  +i[-cos(x)sin(y)+sin(x)cos(y) ]    }{1}\\~\\
 =cos(x-y)+sin(x-y)
  
 \frac{e^{ix}}{e^{iy}}\\
 =e^{ (ix-iy) }\\
 =e^{ i(x-y)   }\\
 =cos(x-y)+isin(x-y)