Simplify (csc(x)-1)(csc(x)+1)

Problem

(csc(x)−1)*(csc(x)+1)

Solution

1. Identify the expression as a difference of squares pattern, which follows the algebraic identity (a−b)*(a+b)=a2−b2

2. Apply the formula by letting a=csc(x) and b=1

(csc(x)−1)*(csc(x)+1)=csc2(x)−1

3. Simplify the constant term.

csc2(x)−1

4. Apply the Pythagorean identity cot2(x)+1=csc2(x) which can be rearranged to csc2(x)−1=cot2(x)

cot2(x)

Final Answer

(csc(x)−1)*(csc(x)+1)=cot2(x)

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