If \frac{1}{\csc \theta-\cot \theta}-\frac{1}{\sin \theta}=x, then what is \frac{1}{\csc \theta+\cot \theta}-\frac{1}{\sin \theta} equal to, where 0 \lt \theta \lt \frac{\pi}{2}?

Explanation

Using \csc^2\theta - \cot^2\theta = 1, we have \frac{1}{\csc\theta-\cot\theta} = \csc\theta+\cot\theta. Thus x = \csc\theta+\cot\theta-\csc\theta = \cot\theta. The required expression is \frac{1}{\csc\theta+\cot\theta} - \frac{1}{\sin\theta} = \csc\theta-\cot\theta-\csc\theta = -\cot\theta = -x.

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