Official solution
A.M. \[ \ge \] G.M.
\[{{{{\sin }^4}\alpha + 4{{\cos }^4}\beta + 1 + 1} \over 4} \ge {\left( {{{\sin }^4}\alpha .4{{\cos }^4}\beta .1.1} \right)^{{1 \over 4}}}\]
sin4\[\alpha \] + 4 cos2\[\beta \] + 2 \[ \ge \] 4 \[\sqrt 2 \] sin \[\alpha \] cos \[\beta \]
Given that sin4\[\alpha \] + 4cos4\[\beta \] + 2 = 4\[\sqrt 2 \] sin\[\alpha \] cos\[\beta \]
\[ \Rightarrow \] A.M.=G.M. \[ \Rightarrow \] sin4\[\alpha \] = 1 = 4 cos4 \[\beta \]
sin \[\alpha \] = 1, cos \[\beta \] = \[ \pm \] \[{1 \over {\sqrt 2 }}\]
\[ \Rightarrow \] sin\[\beta \] = \[{1 \over {\sqrt 2 }}\] as \[\beta \] \[ \in \] [0, \[\pi \]]
cos(\[\alpha \] + \[\beta \]) - cos (\[\alpha \] - \[\beta \]) = - 2 sin \[\alpha \] \[\beta \]
= \[ - \sqrt 2 \]