<s>[INST] Can someone help me simplify this trig expression? 
$$( \tan x+ \sec x )( \cot x-\cos x ) $$

I got stuck after a few steps of converting and adding and what not.
 [/INST] A simplification:
$$
\begin{align*}
\left( \tan x + \sec x \right) \left( \cot x - \cos x  \right) &= \left(\tan x + \frac{1}{\cos x} \right)\left( \frac{1}{\tan x } - \cos x\right) \\
&=\frac{\tan x}{\tan x}-\cos x \tan x +\frac{1}{\cos x \tan x}-\frac{\cos x}{\cos x} \\
&=1-\cos x \tan x +\frac{1}{\cos x \tan x}-1 \\
&= \frac{1}{\cos x \tan x}-\cos x \tan x \\
&= \frac{\cos x}{\cos x \sin x}-\frac{\cos x \sin x}{\cos x} \\
&= \frac{1}{\sin x}-\sin x \\
&= \csc x - \sin x.
\end{align*}
$$
</s>