If ( - + 1 + - 1 ) = 1 k, then k = . . . . . . . .

Answer: Option A

$$\begin{array}{rl}& \left(\frac{\tan \theta -\sec \theta +1}{\tan \theta +\sec \theta -1}\right)\sec \theta =\frac{1}{k}\\ & \left(\frac{\tan \theta -\sec \theta +1}{\frac{1}{\sec \theta -\tan \theta }-1}\right)\sec \theta =\frac{1}{k}\\ & \frac{1}{k}=\left(\frac{\tan \theta -\sec \theta +1}{1-\sec \theta +\tan \theta }\right)\left(\sec \theta -\tan \theta \right)\sec \theta \\ & \frac{1}{k}={\sec }^2\theta -\tan \theta \sec \theta \\ & \frac{1}{k}=\frac{1}{{\cos }^2\theta }-\frac{\sin \theta }{{\cos }^2\theta }\\ & k=\frac{{\cos }^2\theta }{\left(1-\sin \theta \right)}\frac{\left(1+\sin \theta \right)}{\left(1+\sin \theta \right)}\\ & k=\frac{{\cos }^2\theta \left(1+\sin \theta \right)}{{\cos }^2\theta }\\ & k=1+\sin \theta \end{array}$$