Question

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

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

Answer 1

Answer: (B)

$\because$ $x=\sum _{n=0}^{\infty }{\cos }^{2n}\varphi =1+{\cos }^2\varphi +{\cos }^4\varphi +....$
$=\frac{1}{1-{\cos }^2\varphi }=\frac{1}{{\sin }^2\varphi }$
Similarity $=\frac{1}{1-{\sin }^2\varphi }=\frac{1}{{\cos }^2\varphi }$
and $z=\frac{1}{1-{\sin }^2\varphi {\cos }^2\varphi }=\frac{1}{1-\frac{1}{x}.\frac{1}{y}}$
$=\frac{xy}{xy-1}$
$\Rightarrow$ $xyz-z=xy$
$\Rightarrow$ $xyz=xy+z$