Simplify (1+cot(x))(1-cot(x))-csc(x)^2

Problem

(1+cot(x))*(1−cot(x))−csc2(x)

Solution

1. Expand the product of the two binomials using the difference of squares formula (a+b)*(a−b)=a2−b2

1−cot2(x)−csc2(x)

2. Simplify the constant term.

1−cot2(x)−csc2(x)

3. Apply the Pythagorean identity csc2(x)=1+cot2(x) by substituting it into the expression.

1−cot2(x)−(1+cot2(x))

4. Distribute the negative sign through the parentheses.

1−cot2(x)−1−cot2(x)

5. Combine like terms to reach the final simplified form.

−2*cot2(x)

Final Answer

(1+cot(x))*(1−cot(x))−csc2(x)=−2*cot2(x)

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