a) \(A = {(\sin \alpha + \cos \alpha )^2} + {(\sin \alpha - \cos \alpha )^2}\)
\( = 1 + 2\sin \alpha \cos \alpha + 1 - 2\sin \alpha \cos \alpha \)
\( = 2\)
b) \(B = {\sin ^4}\alpha - {\cos ^4}\alpha - 2{\sin ^2}\alpha + 1\)
\( = ({\sin ^2}\alpha + {\cos ^2}\alpha )({\sin ^2}\alpha - {\cos ^2}\alpha )\)\( - 2{\sin ^2}\alpha + 1\)
\( = 1.\left[ {{{\sin }^2}\alpha - \left( {1 - {{\sin }^2}\alpha } \right)} \right] - 2{\sin ^2}\alpha + 1\)
\( = {\sin ^2}\alpha - 1 + {\sin ^2}\alpha - 2{\sin ^2}\alpha + 1\) \( = 0\).
