Complex Number Components Summation

Let S be the set of all (α,β),π<α,β<2π, for which the complex number ‌

1−i‌sin‌α

1+2i‌sin‌α

is purely imaginary and ‌

1+i‌cos‌β

1−2i‌cos‌β

is purely real. Let Zαβ=sin‌2‌α+i‌cos‌2‌β,(α,β)∈S. Then

∑

(α,β)∈S

(iZαβ+‌

1

iZαβ

) is equal to :