Question

For $0<\varphi <\frac{\pi }{2}$, if $x={\sum }_{n=0}^{\infty }{\cos }^{2n}\varphi ,y={\sum }_{n=0}^{\infty }{\sin }^{2n}\varphi$, and $z={\sum }_{n=0}^{\infty }{\cos }^{2n}\varphi {\sin }^{2n}\varphi$, then xyz =

• A. $xy+z$
• B. $xz+y$
• C. $x+y+z$
• D. $yz+x$

Answer 1

Answer: (A), (C)

The correct options are
A $xy+z$
C $x+y+z$

$x={\sum }_{n=0}^{\infty }co{s}^{2n}\varphi =cose{c}^2\varphi$ Sum of infinite terms of GP with common ratio = $co{s}^2\varphi$

$y={\sum }_{n=0}^{\infty }si{n}^{2n}\varphi =se{c}^2\varphi$ Sum of infinite terms of GP with common ratio = $si{n}^2\varphi$
$z={\sum }_{n=0}^{\infty }co{s}^{2n}\varphi si{n}^{2n}\varphi =\frac{1}{1-co{s}^2\varphi si{n}^2\varphi }=\frac{1}{1-\frac{1}{xy}}=\frac{xy}{xy-1}$
so $xyz=xy+z$
or $xyz=x+y+z$ as $xy\Rightarrow x+y$.
Option A and C.