$$2\cos x \leqslant \sqrt{1+\sin 2x} - \sqrt{1-\sin 2x}$$
$$2\cos x \leqslant \sqrt{1+\frac{2\tan x}{1+\tan ^2x}} - \sqrt{1-\frac{2\tan x}{1+\tan ^2x}} $$
$$2\cos x \leqslant \sqrt{\frac{\tan ^2x+2\tan x+1}{1+\tan ^2x}} - \sqrt{\frac{\tan ^2x-2\tan x+1}{1+\tan ^2x}} $$
$$2\cos x \leqslant \frac{1}{\sqrt{1+\tan ^2x}}(|\tan x+1|-|\tan x-1|)$$
$$\leqslant \frac{1}{|\sec x|}(|\tan x+1-\tan x+1|)$$
$$2\cos x \leqslant |2\cos x|$$
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