I've been simplying a slew of trigonometric expressions and most of them fall out pretty clearly, but this one has been giving me fits:
$$\frac{(\sec x - \tan x)^{2} + 1}{\sec x\csc x - \tan x\csc x}$$
Here's my best attempt so far:
$$\begin{align*} &=\frac{\left(\frac{1}{\cos x}-\frac{\sin x}{\cos x} \right)^{2} + 1}{\frac{1}{\cos x\sin x}-\frac{1}{\cos x}}\\\\ &=\frac{\frac{1}{\cos^{2}x}-2\frac{\sin x}{\cos^{2}x}+\frac{\sin^{2}x}{\cos^{2}x} + 1}{\frac{\cos x-\cos x\sin x}{\cos^{2}x\sin x}} \end{align*}$$
But I can't help but think I've already gone wrong since I feel like I should start factoring things out again. Am I on the right track here?
Thanks for any suggestions.