Q. If $0 < \phi < \frac{\pi}{2},\ x = \sum_{n = 0}^{\infty}{\cos^{2n}\phi},\ y = \sum_{n = 0}^{\infty}{\sin^{2n}\phi}$ and $z = \sum_{n = 0}^{\infty}{\cos^{2n}\phi\sin^{2n}\phi}$, then
A) $xyz = xz + y$
B) $xyz = xy + z$
C) $xyz = x + y + z$
D) $xyz = yz + x$
Explanation
(b)
Since, $x = \sum_{n = 0}^{\infty}{\cos^{2n}\phi}$
$= 1 + \cos^{2}\phi + \cos^{4}\phi + ...$
$= \frac{1}{1 - \cos^{2}\phi} = \frac{1}{\sin^{2}\phi}$ $\lbrack\because\left| \cos x \right| < 1\rbrack$
Similarly, $y = \frac{1}{1 - \sin^{2}\phi} = \frac{1}{\cos^{2}\phi}$
and $z = \frac{1}{1 - \sin^{2}\phi\cos^{2}\phi}$
$= \frac{1}{1 - \frac{1}{x}.\frac{1}{y}} = \frac{xy}{xy - 1}$
$\Rightarrow xyz = xy + z$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.