Question

Prove that (d)/(dx)(csc(x))=-csc(x)cot(x) csc(x)

Answer 1

To prove that (d/dx)(csc(x)) = -csc(x)cot(x), we can differentiate csc(x) using the quotient rule.

Step-by-step explanation:

To prove that (d/dx)(csc(x)) = -csc(x)cot(x), we'll start with the formula for the derivative of the cosecant function:

(d/dx)(csc(x)) = -csc(x)cot(x)

To show this, we'll use the quotient rule:

1. First, differentiate the numerator: d/dx(1) = 0
2. Next, differentiate the denominator: d/dx(sin(x)) = cos(x)
3. Apply the quotient rule: (d/dx)(csc(x)) = (0 * sin(x) - 1 * cos(x)) / (sin^2(x))
4. Simplify the expression: -cos(x) / (sin^2(x))
5. Recall that cot(x) = cos(x) / sin(x), so we can rewrite the expression as: -cos(x) / (sin^2(x)) = -cot(x) * csc(x)

Therefore, we have proven that (d/dx)(csc(x)) = -csc(x)cot(x).