Question

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

• A. $x y z=x z+y$
• B. $x y z=x y+z$
• C. $x y z=x+y+z$
• D. $x y z=y z+x$

Answer 1

Answer: (B)

Since, $x=\displaystyle\sum_{n=0}^{\infty} \cos ^{2 n} \phi$
$=1+\cos ^2 \phi+\cos ^4 \phi+\ldots$
$=\frac{1}{1-\cos ^2 \phi}=\frac{1}{\sin ^2 \phi} (\because|\cos x|<1)$
Similarly, $ y=\frac{1}{1-\sin ^2 \phi}=\frac{1}{\cos ^2 \phi}$
and $ z=\frac{1}{1-\sin ^2 \phi \cos ^2 \phi}=\frac{1}{1-\frac{1}{x} \cdot \frac{1}{y}}=\frac{x y}{x y-1}$
$\Rightarrow x y z=x y+z$