scribd.com/doc/87930409/18/Leads-for-tied-down-random-walk [Theorem 3.7, page 38]

Could you explain me the last equality in the proof? I mean this:

$$\frac{2[\sqrt{1 - s^2t^2} - \sqrt{1-t^2}]}{t^2(1-s^2)} = \sum_{n=0}^{\infty} t^{2n} P(S_{2n} = 0) \left(\frac{1 - s^{2n+2}}{(n+1)(1-s^2)}\right)$$

Thank you very much in advance.