My attempt
$$\log_{\cos{x}}(1+2\cos{x})+\log_{\cos{x}}(1+\cos{x})>1$$
If $~~0<\cos{x}<1~~$both logarithms are valid. Rewriting the sum as a product, and notice that $\log_{\cos{x}}\cos{x}=1~$ I have
$$\log_{\cos{x}}(2\cos^2{x+3\cos{x}+1})>\log_{\cos{x}}\cos{x}$$
$$ 2\cos^2{x+3\cos{x}+1}<\cos{x}$$
$$2\cos^2{x}+2\cos{x}+1<0$$
But $~2\cos^2{x}+2\cos{x}+1~$ is always $>0$.
This inequation have solutions, what did I do wrong?