Question

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

• A. $x y-z=(x+y) z$
• B. $x y+y z+z x=z$
• C. $x y z=4$
• D. $x y+z=(x+y) z$

Answer 1

Answer: (D)

$x=\frac{1}{1-\cos ^{2} \theta}$
$ \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, $1-\frac{1}{x} \times \frac{1}{y}=\frac{1}{z} \Rightarrow z(x y-1)=x y \ldots (1)$
Also, $\frac{1}{x}+\frac{1}{y}=1 $
$\Rightarrow x+y=x y \dots (2)$
From $(i)$ and $(ii) $
$xy + z = xyz =( x + y ) z$