LHS=(1+cos(pi/10))(1+cos((3pi)/10))(1+cos((7pi)/10))(1+cos((9pi)/10))
Puttingvpi/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