Q={{Cos[θ], -Sin[θ]},{Sin[θ], Cos[θ]}};
Simplify[Q.Transpose[Q]]
A= Array[a,{2,2}];
A=(({{α+β, χ}, {χ, α-β}}));
ANew=Simplify[Q.A.Transpose[Q]];
ANew[[2,1]]
FullSimplify[Cos[θ]/.Solve[ANew[[2,1]]==0,θ],
{β>0}]
FullSimplify[Sin[θ]/.Solve[ANew[[2,1]]==0,θ],
{β>0}]

TableForm[
Transpose[
{FullSimplify[ANew[[1,1]]/.Solve[ANew[[2,1]]==0,θ],
{β>0,χ>0}],
FullSimplify[ANew[[2,2]]/.Solve[ANew[[2,1]]==0,θ],
{β>0,χ>0}],
FullSimplify[ANew[[1,1]]/.Solve[ANew[[2,1]]==0,θ],
{β>0,χ<0}],
FullSimplify[ANew[[2,2]]/.Solve[ANew[[2,1]]==0,θ],
{β>0,χ<0}]}
],
TableHeadings->{Automatic,{"1,1 χ>0","2,2 χ<0","1,1 χ<0","2,2 χ<0"}}
]