Hi Rahul,
Upon simplifying, you obtain $$-\dfrac{\left(\log x\ -\ \log y\right)^2}{\log x\ \times\ \log y}$$ not $$-\dfrac{1}{\log x\ \times\ \log y}$$. Please recalculate and send us your calculation if you are still obtaining the value to be $$-\dfrac{1}{\log x\ \times\ \log y}$$ so we can check and get back to you with your mistakes if you have committed any.
Hope this helps!

$$\frac{\log_x\left(\frac{x}{y}\right)}{\log_y\left(\frac{y}{x}\right)}=\frac{\left(1-\log_xy\right)}{\left(1-\log_yx\right)}=\frac{\left(1-z\right)}{\left(1-\frac{1}{z}\right)}=-z=\log_x\left(\frac{1}{y}\right)$$

So we finally end up with the expression of $$\log_x\left(\frac{1}{y}\right)\ $$ as the final expression. 
Now, x cannot equal 1 since it goes against log rules and the only way x=1/y for the expression to equal one, is if one of them is less than zero, which is not possible. Hence 1 is a value that is not possible.