ANSWER
By simplifying the left hand side using the Pythagorean Identity.
EXPLANATION
The given identity is
[tex] \csc^{2} (x) - \cot^{2} (x) = 1[/tex]
Take the left hand side and simplify to get the right hand side.
[tex] \csc^{2} (x) - \cot^{2} (x) = \frac{1}{\sin^{2} (x)} - \frac{ \cos^{2} (x)}{\sin^{2} (x)} [/tex]
Collect LCM for the denominators.
[tex]\csc^{2} (x) - \cot^{2} (x) = \frac{1 - \cos^{2} (x)}{\sin^{2} (x)} [/tex]
Recall the Pythagorean Identity.
[tex] \cos^{2} (x) + \sin^{2} (x) = 1[/tex]
This implies that:
[tex]1 - \cos^{2} (x) = \sin^{2} (x)[/tex]
We substitute this to get,
[tex]\csc^{2} (x) - \cot^{2} (x) = \frac{\sin^{2} (x)}{\sin^{2} (x)} [/tex]
[tex]\csc^{2} (x) - \cot^{2} (x) = 1[/tex]
