We know that, #sin((7pi)/14)=sin(pi/2)=1# and take,
X=#sin(pi/14)sin((3pi)/14)sin((5pi)/14)sin((9pi)/14)sin((11pi)/14)sin
((13pi)/14)# Using #(1)# above ,we get
#sin((3pi)/14)=cos(pi/2-(3pi)/14)=cos((4pi)/14)=cos((2pi)/7)#
#sin((5pi)/14)=cos(pi/2-(5pi)/14)=cos((2pi)/14)=cos(pi/7)#
#sin((9pi)/14)=cos(pi/2-(9pi)/14)=cos((-2pi)/14)=cos((pi)/7)#
#sin((11pi)/14)=cos(pi/2-(11pi)/14)=cos((-4pi)/14)=cos((2pi)/7)#
#sin((13pi)/14)=sin((14pi-pi)/14)=sin(pi-pi/14)=sin(pi/14)#
So,
#X=sin(pi/14)cos((2pi)/7)cos(pi/7)cos(pi/7)cos((2pi)/7)sin(pi/14)#
#=[sin(pi/14)cos(pi/7)cos((2pi)/7)]^2#
Applying #(4)# several times,we get
#=
[1/(2cos(pi/14))xx{2sin(pi/14)cos(pi/14)}cos(pi/7)cos((2pi)/7)]^2#
#=[1/(2cos(pi/14))xx{sin(pi/7)}cos(pi/7)cos((2pi)/7)]^2#
#=[1/(4cos(pi/14))xx{2sin(pi/7)cos(pi/7)}cos((2pi)/7)]^2#
#=[1/(4cos(pi/14))xx{sin((2pi)/7)}cos((2pi)/7)]^2#
#=[1/(8cos(pi/14))xx2sin((2pi)/7)cos((2pi)/7)]^2#
#=[1/(8cos(pi/14))xxsin((4pi)/7)]^2#
#=[1/(8cos(pi/14))xxcos(pi/2-(4pi)/7)]^2#
#=[1/cancel(8cos(pi/14))xxcancelcos((pi)/14)]^2#
#X=1/64#