Given that $\sin \theta \ne \pm 1$, prove the identity

$$\frac{1}{1-\sin \theta }+\frac{1}{1+\sin \theta }\equiv 2{\sec }^2\theta$$

Hence, find the set of values of $k$ for which the equation

$$\frac{1}{1-\sin \theta }+\frac{1}{1+\sin \theta }=k$$

has real solutions. Fully justify your answer.

Given that $\theta$ is in the second quadrant (reflex angle between $90^{\circ }$ and $180^{\circ }$) and

$$\frac{1}{1-\sin \theta }+\frac{1}{1+\sin \theta }=10$$

find the exact value of $\tan \theta$.