#cos^2theta+2sin^2theta+3sin^2theta+4sin^2theta+....+# upto 200 terms = 10025
#cos^2theta=1-sin^2theta#
Thus,
#1-sin^2theta+2sin^2theta+3sin^2theta+4sin^2theta+...=10025#
#1-2sin^2theta+sin^2theta+2sin^2theta+3sin^2theta+4sin^2theta+....=10025#
#=(1-2sin^2theta)+sin^2theta(1+2+3+4+.....#upto#200 terms)=10025#
#200th# term #=200#
Hence, the sum happens to be
#(1-2sin^2theta)+sin^2theta(1+2+3+4+....200)=10025#
#(1-2sin^2theta)+sin^2theta(200 )( 201/2)=10025#
#(1-2sin^2theta)+sin^2theta(100*201)=10025#
#(1-2sin^2theta)+sin^2theta(20100)=10025#
#(1-2sin^2theta)+20100sin^2theta=10025#
#1-2sin^2theta+20100sin^2theta=10025#
#20098sin^2theta=10025-1#
#20098sin^2theta=10024#
#sin^2theta=10024/20098#
#sintheta=sqrt((10024/20098)#
#sintheta=0.706#
#cos^2theta=1-sin^2theta#
#cos^2theta=1-10024/20098#
#costheta=0.708#
#sintheta-costheta=0.706-0.708#
#sintheta-costheta=-0.002#
