Let α,  β  be such that π<α−β<3π  If sinα+sinβ=−2165  and cosα+cosβ=−2765   then cos(α−β2)=

π<α−β<3π⇒π2<α−β2<3π2 (sinα+sinβ)2+(cosα+cosβ)2=212652+2726522+2cos(α−β)=9652(49+81)=965×65×130=18651+cos(α−β)=965⇒2cos2(α−β2)=965⇒cos2(α−β2)=9130cos(α−β2)=−9130=−3130