For $0

Question

For $0<\phi<\pi / 2$, if $x=\sum_{n=0}^{\infty} \cos ^{2 n} \phi, y=\sum_{n=0}^{\infty} \sin ^{2 n} \phi$, and $z=\sum_{n=0}^{\infty} \cos ^{2 n} \phi \sin ^{2 n} \phi$, then $x y z=$

Professor PC Thomas Classes · Answer

$ \begin{aligned} & x=\sum_{n=0}^x \cos ^{2 n} \phi=\cos 	heta c^2 \phi \ & \qquad y=\sum_{n=0}^{\infty} \sin ^{2 n} \phi=\sec ^2 \phi \ & z=\sum_{n=0}^{\infty} \cos ^{2 n} \phi \sin ^{2 n} \phi=\frac{1}{1-\cos ^2 \phi \sin ^2 \phi}=\frac{1}{1-\frac{1}{x y}}=\frac{x y}{x y-1} \ & 	ext { so } x y z=x y+z 	ext { or } x y z=x+y+z 	ext { as } x y \Rightarrow x+y \end{aligned} $