Mathematics - Sequence & Series Question with Solution | TestHub

$y=1+{\cos }^2\theta +{\cos }^4\theta +\dots$   

Above series is an infinite G.P, apply sum of infinite G.P.

$\Rightarrow y=\frac{1}{1-{\cos }^2\theta }$

$\Rightarrow \frac{1}{y}={\sin }^2\theta$

$x=\frac{1}{1-\left(-{\tan }^2\theta \right)}=\frac{1}{{\sec }^2\theta }$

$\Rightarrow {\cos }^2\theta =x$

$\Rightarrow \frac{1}{y}+x=1$

$\Rightarrow y\left(1-x\right)=1$