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

Question

asked Apr 1, 2020 39.9k views

1 Answer

Sandoz · Answer 1 · 2020-04-05T21:29:45+0000

ANSWER

By simplifying the left hand side using the Pythagorean Identity.

Step-by-step 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.

$\csc^(2) (x) - \cot^(2) (x) = (1)/(\sin^(2) (x)) - ( \cos^(2) (x))/(\sin^(2) (x))$

Collect LCM for the denominators.

$\csc^(2) (x) - \cot^(2) (x) = (1 - \cos^(2) (x))/(\sin^(2) (x))$

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,

$\csc^(2) (x) - \cot^(2) (x) = (\sin^(2) (x))/(\sin^(2) (x))$

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