Factor the trigonometric expression. There is more than one correct form of the answer.

cot^2 x + csc x − 19

NOT SURE WHETHER TO REPLACE THE COT WITH COS^2X/SIN^2X THEN CSCX TO 1/SINX?

cot^2 = csc^2 - 1

use that and it factors easily.

To factor the trigonometric expression cot^2(x) + csc(x) - 19, we can start by using the trigonometric identities.

First, we can rewrite cot^2(x) as cos^2(x) / sin^2(x) and csc(x) as 1 / sin(x). This substitution helps us simplify the expression.

So the expression becomes:
cos^2(x) / sin^2(x) + 1 / sin(x) - 19

Now, we can find a common denominator for the terms. The common denominator will be sin^2(x).

The expression becomes:
(cos^2(x) + sin(x) - 19 * sin^2(x)) / sin^2(x)

Now, we can factor out sin(x) from the numerator:
sin(x) * (cos^2(x) / sin(x) + 1 - 19 * sin(x)) / sin^2(x)

Simplifying further, we get:
sin(x) * (cos^2(x) + sin(x) - 19 * sin^2(x)) / sin^2(x)

So one possible factored form of the expression is:
sin(x) * (cos^2(x) + sin(x) - 19 * sin^2(x)) / sin^2(x)

Alternatively, if we want to simplify it further, we could factor out sin(x) from both the numerator and the denominator:
(sin(x) * (cos^2(x) + sin(x) - 19 * sin^2(x))) / (sin(x) * sin(x))

Which can be further simplified to:
(sin(x) * (cos^2(x) + sin(x) - 19 * sin^2(x))) / sin^2(x)

So, there are two possible forms of the factored expression:
1) sin(x) * (cos^2(x) + sin(x) - 19 * sin^2(x)) / sin^2(x)
2) (sin(x) * (cos^2(x) + sin(x) - 19 * sin^2(x))) / sin^2(x)