The correct option is A k∈[nπ+π3,nπ+2π3]
cosx+cos(k+x)−cos(k−x)=2⇒cosx−2 sink⋅sinx=2
We know that for an equation acosx+bsinx=c to have solution |c|≤√a2+b2
√1+4sin2k≥2⇒1+4sin2k≥4⇒|sink|≥√32⇒sink≥√32 or sink≤−√32⇒k∈[2nπ+π3,2nπ+2π3] or k∈[(2n+1)π+π3,(2n+1)π+2π3]⇒k∈[nπ+π3,nπ+2π3]