#LHS=(1+cos(pi/10))(1+cos((3pi)/10))(1+cos((7pi)/10))(1+cos((9pi)/10))#
Puttingv#pi/10=2theta# we have
#LHS=(1+cos(2theta))(1+cos(6theta))(1+cos(14theta))(1+cos(18theta))#
#=(2cos^2(theta))(2cos^2(3theta))(2cos^2(7theta))(2cos^2(9theta))#
#=[(2cos(theta)cos(9theta))(2cos(3theta)cos(7theta))]^2#
#=[(cos(10theta)+cos(8theta))(cos(10theta)+cos(4theta))]^2#
#=[(cos(pi/2)+cos(8theta))(cos(pi/2)+cos(4theta))]^2#
#=[cos(8theta)cos(4theta)]^2#
#=[(4cos(8theta)cos(4theta)sin(4theta))/(4sin(4theta))]^2#
#=1/16[sin(16theta)/sin(4theta)]^2#
#=1/16[sin((8 pi)/10)/sin((2pi)/10)]^2#
#=1/16[sin(pi-(2pi)/10)/sin((2pi)/10)]^2#
#=1/16[sin((2pi)/10)/sin((2pi)/10)]^2#
#=(1/16)=RHS#