sequences and series - Find the sum : $sin^{-1}frac{1}{sqrt{2}}+sin^{-1}frac{sqrt{2}-1}{sqrt{6}}+sin^{-1}frac{sqrt{3}-sqrt{2}}{sqrt{12}}+cdots$

Problem :

Find the sum of :

$$\sin^{-1}\frac{1}{\sqrt{2}}+\sin^{-1}\frac{\sqrt{2}-1}{\sqrt{6}}+\sin^{-1}\frac{\sqrt{3}-\sqrt{2}}{\sqrt{12}}+\cdots$$

My approach :

Here the $n$'th term is given by :

$$t_n = \sin^{-1}\left[\frac{\sqrt{n}-\sqrt{n-1}}{\sqrt{n}\sqrt{n+1}}\right]$$

From now how to proceed further please suggest thanks....