Question:
If $0<\theta, \phi<\frac{\pi}{2}, x=\sum_{n=0}^{\infty} \cos ^{2 n} \theta, y=\sum_{n=0}^{\infty} \sin ^{2 n} \phi$
and $z=\sum_{n=0}^{\infty} \cos ^{2 n} \theta \cdot \sin ^{2 n} \phi$ then :
Correct Option: , 4
Solution:
$x=\frac{1}{1-\cos ^{2} \theta} \Rightarrow \sin ^{2} \theta=\frac{1}{x}$
Also, $\cos ^{2} \theta=\frac{1}{y} \& 1-\sin ^{2} \theta \cos ^{2} \theta=\frac{1}{z}$
So, $1-\frac{1}{x} \times \frac{1}{y}=\frac{1}{z} \Rightarrow z(x y-1)=x y$.....(1)
Also, $\frac{1}{x}+\frac{1}{y}=1 \quad \Rightarrow x+y=x y$.........(2)
From (i) and (ii)
$x y+z=x y z=(x+y) z$
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