Prove that `tan^(-1).(1)/(sqrt2) + sin^(-1).(1)/(sqrt5) - cos^(-1).(1)/(sqrt10) = -pi + cot^(-1) ((1 + sqrt2)/(1 - sqrt2))`

Explanation

To prove the equation

\[ \tan^{-1}\left(\frac{1}{\sqrt{2}}\right) + \sin^{-1}\left(\frac{1}{\sqrt{5}}\right) - \cos^{-1}\left(\frac{1}{\sqrt{10}}\right) = -\pi + \cot^{-1}\left(\frac{1 + \sqrt{2}}{1 - \sqrt{2}}\right), \]

we will convert \(\sin^{-1}\) and \(\cos^{-1}\) into terms of \(\tan^{-1}\) and then simplify the left-hand side.

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