Question number 4 please! Question number 4 please! equal to (c) 10 cos— + i sin — 2 is (cosO+i sin Z/" • (sin0+icos0)5 (d) 23 _ i) +14i = O, then z i IS 9. (d) 40 cos— + i sin — 4 4 cos—+i sin— ...T 8 8 (d) None of these is equal to Share with your friends Share 0 Vijay Kumar Gupta answered this Dear Student, Consider the following product. cosπ2+isinπ2 cosπ4+isinπ4 cosπ8+isinπ8 ⋯⋯ ∞Let us suppose that, cosπ2+isinπ2=eiπ2Then, we have cosπ4+isinπ4=eiπ4=eiπ22 cosπ8+isinπ8=eiπ8=eiπ23 ⋮ ⋮Therefore, the given product series becomes, cosπ4+isinπ4 cosπ8+isinπ8 ⋯⋯ ∞ =eiπ2 · eiπ22 · eiπ23 ⋯⋯ ∞ =eiπ2+iπ22+iπ23+⋯⋯∞ =eiπ2 1+12+122+⋯⋯∞ ..... 1Note that the series 1+12+122+⋯⋯∞ is an infinite geometricseries and its sum is given by, 1+12+122+⋯⋯∞=11-12 =112=2Thus from 1, we get cosπ4+isinπ4 cosπ8+isinπ8 ⋯⋯ ∞ =eiπ2 2 =eiπ =cosπ+isinπ =-1+i0 =-1 Regards 0 View Full Answer
