I am given the dynamical system $$x_{n+1}=\frac{1}{2}\bigg(x_n-\frac{1}{x_n}\bigg) \quad \quad \text{ for } n=0,1,2,...$$
I'm then told to substitute $x_n=cot \ y_n$ and find a difference equation for $y_n.$
Next I need to find the solution $x_n$ in terms of $n$ and $x_0$.
So far I have made the substitution and found $$\cot \ y_{n+1}=\frac{1}{2}\bigg(\cot \ y_n - \frac{1}{\cot \ y_n}\bigg)$$
Then rearranging this I get $$y_{n+1}=2y_n$$
I gave then got down to $$y_n=y_0 2^{n}$$
Which then gives me $$x_n=\cot(\cot^{-1}(x_0)\cdot 2^{n})$$
Is this right?