Show that the map \(R_{X,\pi}\colon S^2\to S^2\) given by \((a,b,c)\to (a,-b,-c)\) (rotation about the \(x\)-axis by \(\pi\) radians) and the map \(T_{X,\pi}\colon \hat{\mathbb{C}}\to \hat{\mathbb{C}}...Show that the map \(R_{X,\pi}\colon S^2\to S^2\) given by \((a,b,c)\to (a,-b,-c)\) (rotation about the \(x\)-axis by \(\pi\) radians) and the map \(T_{X,\pi}\colon \hat{\mathbb{C}}\to \hat{\mathbb{C}}\) given by \(z\to 1/z\) are conjugate transformations with respect to stereographic projection.
