# If 0 less than theta less than pi/2, x = sigma n = 0 to infinity cos^2n theta, y = sigma n = 0 to infinity sin^2n theta and z = sigma n = 0 to infinity cos^2n theta sin^2n theta, then xyz (1) xyz = xz+y (2) xyz = xy+z (3) xyz = yz+x (4) None of these

If 0<theta < pi/2, x = sigma n = 0 to infinity cos2n theta, y = sigma n = 0 to infinity sin2n theta and z = sigma n = 0 to infinity cos2n theta sin2n theta, then xyz (1) xyz = xz+y (2) xyz = xy+z (3) xyz = yz+x (4) None of these

Solution:

Given x = Σn=0 cos2n θ

= 1+cos2θ+cos4θ+…

= 1/(1-cos2θ) (since a = 1, r = cos2θ)

= 1/sin2θ

y = Σn=0 sin2n θ

= 1+ sin2 θ + sin4 θ+…

= 1/(1-sin2θ)

= 1/cos2θ

z = 1/(1-sin2θcos2θ)

= 1/(1-(1/x)(1/y))

= xy/(xy-1)

(xy-1)z = xy

xyz-z = xy

xyz = xy+z

Hence option (2) is the answer.