If $-\pi /2<x<\pi /2,$ and the sum to infinite terms of the series

$\cos x+\frac{2}{3}\cos x{\sin }^{2}x+\frac{4}{9}\cos x{\sin }^{4}x+\dots$

is finite, then

The given series is an infinite G.P. with common ratio$(2/3)$ $\sin 2x$. For the sum to exist, we must have

$\left|(2/3){\sin }^{2}x\right|<1$

We have

$\left|\frac{2}{3}{\sin }^{2}x\right|=\frac{2}{3}|\sin x{|}^2<\frac{2}{3}<1$

Therefore, the sum of the series exists and is given by

$S=\frac{\cos x}{1-\frac{2}{3}{\sin }^{2}x}=\frac{3\cos x}{2+\cos 2x}$

Clearly, $\cos x and 2 + \cos 2x$ are finite for $-\pi /2<x<\pi /2$

So, the sum of the given series is finite for $x\in (-\pi /2,\pi /2)$.