Simplify (cot(x)+csc(x))(cot(x)-csc(x))

Problem

(cot(x)+csc(x))*(cot(x)−csc(x))

Solution

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

2. Apply the formula by squaring the first term and subtracting the square of the second term.

cot^2(x)−csc^2(x)

3. Use the Pythagorean identity for trigonometry, which states 1+cot^2(x)=csc^2(x)

4. Rearrange the identity to find the value of the expression by subtracting csc^2(x) and 1 from both sides.

cot^2(x)−csc^2(x)=−1

Final Answer

(cot(x)+csc(x))*(cot(x)−csc(x))=−1

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