15. Find the minimum value of
$$
\left|\sin x+\cos x+\frac{\cos x-\sin x}{\cos 2 x}\right|
$$
15. Answer: 2
Note that $\cos 2 x=\cos ^{2} x-\sin ^{2} x$. So
$$
\left|\sin x+\cos x+\frac{\cos x-\sin x}{\cos 2 x}\right|=\left|\sin x+\cos x+\frac{1}{\sin x+\cos x}\right| .
$$

Set $w=\sin x+\cos x$ and minimize $\left|w+\frac{1}{w}\right|$.
By AM-GM inequality, if $w$ is positive then the minimum of $w+\frac{1}{w}$ is 2 ; if $w$ is negative, then the maximum of $w+\frac{1}{w}$ is -2 . Therefore, the minimum of $\left|w+\frac{1}{w}\right|$ is 2.