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)