Prove the following
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

  1. (1) $x y z=4$

  2. (2) $x y-z=(x+y) z$

  3. (3) $x y+y z+z x=z$

  4. (4) $x y+z=(x+y) z$


Correct Option: , 4

Solution:

$x=1+\cos ^{2} \theta+\ldots \ldots \ldots \infty$

$x=\frac{1}{1-\cos ^{2} \theta}=\frac{1}{\sin ^{2} \theta}$

………

$y=1+\sin ^{2} \phi+\ldots \ldots \ldots$

$y=\frac{1}{1-\sin ^{2} \phi}=\frac{1}{\cos ^{2} \phi}$

$z=\frac{1}{1-\cos ^{2} \theta \cdot \sin ^{2} \phi}=\frac{1}{1-\left(1-\frac{1}{x}\right)\left(1-\frac{1}{y}\right)}=\frac{x y}{x y-(x-1)(y-1)}$

$x z+y z-z=x y$

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

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