Answer
$$\lim_{x\to0}\sin\Big(\frac{\pi+\tan x}{\tan x-2\sec x}\Big)=-1$$
Work Step by Step
$$A=\lim_{x\to0}\sin\Big(\frac{\pi+\tan x}{\tan x-2\sec x}\Big)$$
$$A=\sin\Big(\frac{\pi+\tan 0}{\tan0-2\sec0}\Big)$$
$$A=\sin\Big(\frac{\pi+0}{0-\frac{2}{\cos0}}\Big)$$
$$A=\sin\Big(\frac{\pi}{-\frac{2}{1}}\Big)$$
$$A=\sin\Big(-\frac{\pi}{2}\Big)=-1$$
