Question

How can csc^2-cot^2 = 1?

Answer 1

ANSWER

By simplifying the left hand side using the Pythagorean Identity.

EXPLANATION

The given identity is

$\csc^{2} (x) - \cot^{2} (x) = 1$

Take the left hand side and simplify to get the right hand side.

$\csc^{2} (x) - \cot^{2} (x) = \frac{1}{\sin^{2} (x)} - \frac{ \cos^{2} (x)}{\sin^{2} (x)}$

Collect LCM for the denominators.

$\csc^{2} (x) - \cot^{2} (x) = \frac{1 - \cos^{2} (x)}{\sin^{2} (x)}$

Recall the Pythagorean Identity.

$\cos^{2} (x) + \sin^{2} (x) = 1$

This implies that:

$1 - \cos^{2} (x) = \sin^{2} (x)$

We substitute this to get,

$\csc^{2} (x) - \cot^{2} (x) = \frac{\sin^{2} (x)}{\sin^{2} (x)}$

$\csc^{2} (x) - \cot^{2} (x) = 1$