Angle subtended by the diameter of each of 2003 unit circles (or the distance between the centers of two adjacent unit circles) at the center of the circle of radius #r# will be #angle AOB=((2pi)/2003)#.
The length of the straight line #OA# joining the center #O# of the large circle of radius #r# with any of the center #A# of the unit circle of unit radius will be #OA=r+1# . The length of the straight line joining the point of contact #C#of two adjacent unit circle with the center of the large circle will be perpendicular to #AC#, the radius of the unit circle.
Hence in #Delta ACO, angle ACO=90^@#
Its hypotenuse #OA=r+1# and opposite #AC=1#
This right triangle will have an angle #angleAOC=pi/2003# at the center of the large circle.
For this right triangle we have #cosec(pi/2003)=(OA)/(AC)=(r+1)/1#
Hence #r=cosec(pi/2003)-1#
