Question
$0 < \phi < \frac{\pi }{2}$ के लिये, यदि $x = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\phi ,} $ $y = \sum\limits_{n = 0}^\infty {{{\sin }^{2n}}\phi ,} $ $z = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\phi \,{{\sin }^{2n}}\phi ,} $ तो
$y = 1 + {\sin ^2}\phi + {\sin ^4}\phi + .... = \frac{1}{{(1 - {{\sin }^2}\phi )}} = \frac{1}{{{{\cos }^2}\phi }}$
$z = 1 + {\cos ^2}\phi {\sin ^2}\phi + {\cos ^4}\phi {\sin ^4}\phi + .. = \frac{1}{{(1 - {{\cos }^2}\phi {{\sin }^2}\phi )}}$
अब $xyz = \frac{1}{{{{\sin }^2}\phi {{\cos }^2}\phi (1 - {{\cos }^2}\phi {{\sin }^2}\phi )}}$
$xy + z = \frac{1}{{{{\sin }^2}\phi {{\cos }^2}\phi }} + \frac{1}{{1 - {{\cos }^2}\phi {{\sin }^2}\phi }}$
$ = \frac{1}{{{{\sin }^2}\phi {{\cos }^2}\phi (1 - {{\cos }^2}\phi {{\sin }^2}\phi )}} = xyz$
जो कि $(b)$ में दिया है।
एवं $x + y + z = xyz$, जो $(c)$ में दिया है .
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