Question

`"For "0ltthetalt(pi)/(2)`, if
`x=sum_(n=0)^(oo)cos^(2n)theta,y=sum_(n=0)^(oo)sin^(2n)phi,z=sum_(n=0)^(oo)cos^(2n)thetasin^(2n)phi`, then
A. xy=zx+zy+z
B. xy=zx+zy-z
C. xy+yz+zx=z
D. none of these

Answer 1

Correct Answer - B
We have,
`x=underset(n=0)overset(oo)sumcos^(2n)theta=1+cos^(2)theta+cos^(4)theta+ . . . .oo`
`y=underset(n=0)overset(oo)sumsin^(2n)phi=1+sin^(4)phi+ . . . .oo`
`and,z=underset(n=0)overset(oo)sumcos^(2n)thetasin^(2n)phi`
`rArr" "z=1+cos^(2)thetasin^(2)phi+cos^(4)thetasin^(4)phi+ . . . ..oo`
`:." "x=(1)/(1-cos^(2)theta),y=(1)/(1-sin^(2)phi)andz=(1)/(1-cos^(2)thetasin^(2)phi)`
`rArr" "x=(1)/(sin^(2)theta),y=(1)/(cos^(2)phi)andz=(1)/(1-cos^(2)thetasin^(2)phi)`
`rArr" "(1)/(1-(1-(1)/(x))(1-(1)/(y)))=(xy)/(xy-(xy-x-y+1))`
`rArr" "z=(xy)/(x+y-1)rArrxy=zx+zy-z`