If cos α+cos β=0=sin α+sin β, then prove that cos 2α+cos 2β=−2 cos (α+β)
cos α+cos β=0=sin α+sin β
Squaring on both sides gives
cos2 α+cos2 β+2 cos α cos β=sin2 α+sin2β+2 sin αsin β
Bring square terms on one side, we get
cos 2α+cos 2 β=−2 (−sin α sin β+cos α cos β)=−2 cos (α+β)