I'm really confused about the operation rules of the Big Theta notation. For example, I have met the following derivation process:
$$ cos(\theta_r(y))=1+2(e^{-2y}+e^{-2r}-e^{R-r-y}-e^{-R-r-y})(1+\Theta(e^{-2r}))(1+\Theta(e^{-2y})) = 1-2e^{R-r-y}+\Theta(e^{-2y}+e^{-2r}) $$ The derivation is performed by the use of condition $y\ge R-r$.
Can anyone give me some tips on this derivation step ? How can it come to the second expression ?
Thanks!