If $\cos x+\sec x=-2$ then for a positive integer $n,{\cos }^{n}x+{\sin }^{n}x$

If $\alpha ,\beta$ and $\gamma$ are connected by the relation

$2{\tan }^2\alpha {\tan }^2\beta {\tan }^2\gamma +{\tan }^2\alpha {\tan }^2\beta +{\tan }^2\beta {\tan }^2\gamma +{\tan }^2\gamma {\tan }^2\alpha =1$, then