Question

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

• A. $xyz=xz+y$
• B. $xyz=xy+z$
• C. $xyz=x+y+z$
• D. $x{y}^2=y^2+x$

Answer 1

Answer: (B), (C)

$xyz=xy+z$
$xyz=x+y+z$

Simplifying, we get
$x={cosec}^2\theta$.
$y={\sec }^2\theta$
Hence
$z=\frac{1}{1-{\cos }^2\theta .{\sin }^2\theta }=\frac{1}{1-\frac{1}{xy}}$
$=\frac{xy}{xy-1}$
Or
$xyz-z=xy$
Or
$xyz=xy+z$.
Now
$xy={\sec }^2\theta .{cosec}^2\theta$

$=\frac{1}{{\sin }^2\theta }.\frac{1}{{\cos }^2\theta }$

$=\frac{{\sin }^2\theta +{\cos }^2\theta }{{\cos }^2\theta .{\sin }^2\theta }$

$=\frac{1}{{\cos }^2\theta }+\frac{1}{\sin 2\theta }$

$=x+y$.
Hence
$xyz=x+y+z$.