Prove that matrix `[((b^(2)-a^(2))/(a^(2)+b^(2)),(-2ab)/(a^(2)+b^(2))),((-2ab)/(a^(2)+b^(2)),(a^(2)-b^(2))/(a^(2)+b^(2)))]` is orthogonal.

Explanation

`A=[((b^(2)-a^(2))/(a^(2)+b^(2)),(-2ab)/(a^(2)+b^(2))),((-2ab)/(a^(2)+b^(2)),(a^(2)-b^(2))/(a^(2)+b^(2)))]=[((1-a^(2)/b^(2))/(1+a^(2)/b^(2)),(-2 a/b)/(1+a^(2)/b^(2))),((-2a/b)/(1+a^(2)/b^(2)),-(1-a^(2)/(2))/(1+a^(2)/b^(2)))]`
`=[(cos 2 theta,-sin 2 theta),(-sin 2 theta,-cos 2 theta)]`, where `a/b= tan theta`
`:. A A^(T)=[(cos 2 theta,-sin 2 theta),(-sin 2 theta,-cos 2 theta)][(cos 2 theta,-sin 2 theta),(- sin 2 theta,-cos 2 theta)]`
`=[(1,0),(0,1)]`
Thus, A is orthogonal.