Question:
If $0<\theta, \phi<\frac{\pi}{2}, \mathrm{x}=\sum_{\mathrm{n}=0}^{\infty} \cos ^{2 \mathrm{n}} \theta, \mathrm{y}=\sum_{\mathrm{n}=0}^{\infty} \sin ^{2 \mathrm{n}} \phi$ and
$\mathrm{z}=\sum_{\mathrm{n}=0}^{\infty} \cos ^{2 \mathrm{n}} \theta \cdot \sin ^{2 \mathrm{n}} \phi$ then
Correct Option: , 4
Solution:
$x=1+\cos ^{2} \theta+\ldots \ldots \ldots \infty$
$x=\frac{1}{1-\cos ^{2} \theta}=\frac{1}{\sin ^{2} \theta}$
.........
$y=1+\sin ^{2} \phi+\ldots \ldots \ldots$
$y=\frac{1}{1-\sin ^{2} \phi}=\frac{1}{\cos ^{2} \phi}$
$z=\frac{1}{1-\cos ^{2} \theta \cdot \sin ^{2} \phi}=\frac{1}{1-\left(1-\frac{1}{x}\right)\left(1-\frac{1}{y}\right)}=\frac{x y}{x y-(x-1)(y-1)}$
$x z+y z-z=x y$
$x y+z=(x+y) z$