In this problem I have to ask that horse exerts a force F on ground and reaction R is given by ground. The horizontal component of R i.e. RcosƟ push the horse forward.But I have to ask that this RcosƟ will be the reaction to horizontal component of F i.e. FcosƟ so they both will cancel each other then how RcosƟ will push the cart forward ,or as the whole R is the opposite reaction of F so it will end up in cancelling F then how any of it's component can pull the cart.Please help me as my exams are coming

Think of the horse and cart as a single entity, and consider the forces acting on it. In the horizontal direction only Rcos(theta) is acting, so the horse & cart are accelerated horizonatally according to \Sigma \vec F = m \vec a. Now it's true that magnitude of R cos(theta) is equal to the horizontal force with which the horse pushes against the ground, but thety don't "cancel out."