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(D) Given $x = \sum_{n=0}^{\infty} (\cos^2 	heta)^n = \frac{1}{1 - \cos^2 	heta} = \frac{1}{\sin^2 	heta}$ $\Rightarrow \sin^2 	heta = \frac{1}{x}

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$xy + z = (x + y)z$

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(D) Given $x = \sum_{n=0}^{\infty} (\cos^2 	heta)^n = \frac{1}{1 - \cos^2 	heta} = \frac{1}{\sin^2 	heta}$ $\Rightarrow \sin^2 	heta = \frac{1}{x}$.Similarly,$y = \sum_{n=0}^{\infty} (\sin^2 \phi)^n = \frac{1}{1 - \sin^2 \phi} = \frac{1}{\cos^2 \phi}$ $\Rightarrow \cos^2 \phi = \frac{1}{y}$.And $z = \sum_{n=0}^{\infty} (\cos^2 	heta \sin^2 \phi)^n = \frac{1}{1 - \cos^2 	heta \sin^2 \phi}$ $\Rightarrow 1 - \cos^2 	heta \sin^2 \phi = \frac{1}{z}$.Since $\cos^2 	heta = 1 - \sin^2 	heta = 1 - \frac{1}{x} = \frac{x-1}{x}$ and $\sin^2 \phi = 1 - \cos^2 \phi = 1 - \frac{1}{y} = \frac{y-1}{y}$,we have:$1 - \left(\frac{x-1}{x}ight) \left(\frac{y-1}{y}ight) = \frac{1}{z}$.$1 - \frac{xy - x - y + 1}{xy} = \frac{1}{z}$ $\Rightarrow \frac{xy - xy + x + y - 1}{xy} = \frac{1}{z}$ $\Rightarrow \frac{x + y - 1}{xy} = \frac{1}{z}$.$z(x + y - 1) = xy$ $\Rightarrow z(x + y) - z = xy$ $\Rightarrow xy + z = z(x + y)$.

If $0 < \theta, \phi < \frac{\pi}{2}$,$x = \sum_{n=0}^{\infty} \cos^{2n} \theta$,$y = \sum_{n=0}^{\infty} \sin^{2n} \phi$,and $z = \sum_{n=0}^{\infty} \cos^{2n} \theta \cdot \sin^{2n} \phi$,then:

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