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Prove that $\frac{\sin \theta - \cos \theta+1}{\sin \theta + \cos \theta-1} = \frac{1}{\sec \theta - \tan \theta}$
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$$\begin{aligned}& L.H.S = \frac{\sin \theta - \cos \theta+1}{\sin \theta + \cos \theta-1} \\ & \text{Divide Numerator and Denominator by } \cos \theta. \\ & = \frac{\tan \theta-1+\sec \theta}{\tan \theta+1-\sec \theta} \\ & = \frac{\tan \theta-1+\sec \theta}{(\tan \theta-\sec\theta)+(\sec^2\theta-\tan^2\theta)} \\ & = \frac{\tan \theta-1+\sec \theta}{(\sec \theta-\tan \theta) (\tan \theta+\sec\theta-1)} \\ & = \frac{1}{\sec \theta-\tan \theta} \\ & = R.H.S\end{aligned}$$