Show that cos2π/15. cos4π/15.cos8π/15.cos16π/15=1/16

Kinshuk Sharma , 7 years ago

Grade:11

5 Answers

Nishtha Gahlot

40 Points

7 years ago

cos(2π/15) cos(4π/15) cos(8π/15) cos(14π/15)
= cos(2π/15) cos(4π/15) cos(8π/15) cos(π - π/15)
= cos(2π/15) cos(4π/15) cos(8π/15) * -cos(π/15)
= -cos(π/15) cos(2π/15) cos(4π/15) cos(8π/15)
= -16sin(π/15) cos(π/15) cos(2π/15) cos(4π/15) cos(8π/15) / [ 16 sin(π/15) ]
= -8 * [ 2sin(π/15) cos(π/15) ] cos(2π/15) cos(4π/15) cos(8π/15) / [ 16 sin(π/15) ]
= -8 sin(2π/15) cos(2π/15) cos(4π/15) cos(8π/15) / [ 16 sin(π/15) ]
= -4 * [ 2 sin(2π/15) cos(2π/15) ] cos(4π/15) cos(8π/15) / [ 16 sin(π/15) ]
= -4 sin(4π/15) cos(4π/15) cos(8π/15) / [ 16 sin(π/15) ]
= -2 * [2 sin(4π/15) cos(4π/15) ] cos(8π/15) / [ 16 sin(π/15) ]
= -2 sin(8π/15) cos(8π/15) / [ 16 sin(π/15) ]
= -sin(16π/15) / [ 16 sin(π/15) ]
= -sin(π + π/15) / [ 16 sin(π/15) ]
= - * -sin(π/15) / [ 16 sin(π/15) ]
= 1/16

Nishtha Gahlot

40 Points

7 years ago

sorry,it should be cos(16π/15),in the first line. anyway it wouldn’t make a difference because

cos(π+π/15)=-cos(π/15),which is same as taking cos(14π/15).

Harsh Jain

31 Points

6 years ago

Solving it easily there is a trick for solving the cos multiple series . (ONly for cos multiple series)). When the angles are in gp or the next angle is double of previous and series continues.... It is like this ..{Sin(2 *largest angle)}/{sin (smallest angle)*2^n}Where "n" is number of terms in the series In this case ~~~~{Sin (32π/15)}/sin(2π/15)*2^4{Sin(2π/15)}/sin(2π/15)*161/16

Harsh Jain

31 Points

6 years ago

Solving it easily there is a trick for solving the cos multiple series . (ONly for cos multiple series)). When the angles are in gp or the next angle is double of previous and series continues.... It is like this ..{Sin(2 *largest angle)}/{sin (smallest angle)*2^n}Where "n" is nuhmber of terms in the series In this case ~~~~=={Sin (32π/15)}/sin(2π/15)*2^4={Sin(2π/15)}/sin(2π/15)*16==1/16

ankit singh

askIITians Faculty 614 Points

3 years ago