Matrix elements for non-orthogonal spin states in the x-z plane

plus[θ_]:={Cos[θ/2],Sin[θ/2]}

minus[θ_]:={-Sin[θ/2],Cos[θ/2]}

plus[θa].plus[θb]//TrigReduce

minus[θa].plus[θb]//TrigReduce

minus[θa].minus[θb]//TrigReduce