Smallest positive integral value of $a$ such that ${\sin}^2 x+a\cos x+{a}^2>1+\cos x$ holds for all real $x$

Question

If the inequality $${\sin}^2 x+a\cos x+{a}^2>1+\cos x$$ holds for all $x \in \Bbb R$ then what's the smallest positive integral value of $a$?

Here's my approach to the problem $$\cos^2 x+(1-a)\cos x-a^2<0$$ Let us consider this as a quadratic form respect to $a$.

Applying the quadratic formula $a=\frac{-\cos x\pm\sqrt{5\cos^2 x+4\cos x}}2 $ and substituting $\cos x$ with $1$ and $-1$ we get 3 values of where the graph should touch the x axis $-2,0,1$ How should I proceed now?

Answer 1

Write sin$^2$x=1-cos$^2$x, and factorize the resultant inequation.