Find the Derivative - d/dx y=(csc(x)+cot(x))(csc(x)-cot(x))

Problem

d()/d(x)*[(csc(x)+cot(x))*(csc(x)−cot(x))]

Solution

1. Identify the expression as a difference of squares in the form (a+b)*(a−b)=a^2−b^2

2. Expand the product using this algebraic identity.

y=csc^2(x)−cot^2(x)

3. Apply the Pythagorean identity for trigonometric functions, which states that 1+cot^2(x)=csc^2(x)

4. Simplify the expression by substituting the identity, which shows that csc^2(x)−cot^2(x)=1

y=1

5. Differentiate the constant value with respect to x

d(y)/d(x)=d()/d(x)*1

6. Evaluate the derivative of a constant, which is always 0

d(y)/d(x)=0

Final Answer

d()/d(x)*[(csc(x)+cot(x))*(csc(x)−cot(x))]=0

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