Question

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

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

Answer 1

Answer: (B), (C)

The correct options are
B xyz$=$xy+z
C xyz$=$x+y+z

The series x converges to a value $\frac{1}{1-{\cos }^2\varphi }=cose{c}^2\varphi$
Similarly y converges to a value $se{c}^2\varphi$
In a similar fashion.. z converges to $\frac{cose{c}^2\varphi se{c}^2\varphi }{cose{c}^2\varphi se{c}^2\varphi -1}$
$z=\frac{xy}{xy-1}$
$xy+z=xyz$; Option B.
And, also
$x+y=xy$
$\therefore xyz=x+y+z$; Option C.