Question

Simplify the expression: (csc² x - cot² x)(sin(-x) ⋅ cot(x)) and show that it is equal to -sec x .

a) (1)(sin x)

b) -cos x

c) cot x

d) (sin x)(cos x)

Answer 1

The expression (csc² x - cot² x)(sin(-x) ⋅ cot(x)) simplifies to -sec(x). The correct answer is b) -cos(x).

Step-by-step explanation:

To simplify the expression (csc² x - cot² x)(sin(-x) ⋅ cot(x)), we can use the trigonometric identities. First, let's simplify the expression inside the parentheses. The identity csc² x - cot² x = -1:

(-1)(sin(-x) ⋅ cot(x))

Next, we can simplify sin(-x) and cot(x) using the identities sin(-x) = -sin(x) and cot(x) = 1/tan(x):

(-1)(-sin(x) ⋅ 1/tan(x))

Simplifying further, we have -sin(x)/tan(x). Since tan(x) = sin(x)/cos(x), we can substitute this into the expression:

-sin(x)/(sin(x)/cos(x))

Simplifying the expression further, we get -cos(x)/sin(x), which is equal to -sec(x). Therefore, the answer is option b) -cos(x).