If 0

Question

If $0<	heta,\varphi<\dfrac{\pi}{2}$, $x=\displaystyle\sum_{n=0}^{\infty}\cos^{2n}	heta$, $y=\displaystyle\sum_{n=0}^{\infty}\sin^{2n}\varphi$, and $z=\displaystyle\sum_{n=0}^{\infty}\cos^{2n}	heta\sin^{2n}\varphi$, then:

Accepted Answer

$$x = 1 + {\cos ^2}\theta  + ..........\infty $$

$$x = {1 \over {1 - {{\cos }^2}\theta }} = {1 \over {{{\sin }^2}\theta }}$$ .......(1)

$$y = 1 + {\sin ^2}\phi  + ........\infty $$

$$y = {1 \over {1 - {{\sin }^2}\phi }} = {1 \over {{{\cos }^2}\phi }}$$ ....... (2)

$$z = {1 \over {1 - {{\cos }^2}\theta .{{\sin }^2}\phi }} = {1 \over {1 - \left( {1 - {1 \over x}} \right)\left( {1 - {1 \over y}} \right)}} = {{xy} \over {xy - (x - 1)(y - 1)}}$$

$$ \Rightarrow $$ $xz + yz - z = xy$

$$ \Rightarrow $$ $xy + z = (x + y)z$

Solutions · 1