Question

perform the indicated operation and simplify the result so that there are no quotients. (cotx - 1)^2 (cotx - 1)^2 = (simplify your answer.)

Explanation:

Step1: Expand the square

Use the formula \((a - b)^2=a^{2}-2ab + b^{2}\), where \(a = \cot x\) and \(b = 1\).
\((\cot x-1)^{2}=\cot^{2}x-2\cot x + 1\)

Step2: Use the Pythagorean identity

Recall the Pythagorean identity \(1+\cot^{2}x=\csc^{2}x\), so \(\cot^{2}x=\csc^{2}x - 1\).
Substitute \(\cot^{2}x=\csc^{2}x - 1\) into \(\cot^{2}x-2\cot x + 1\):

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Step3: Simplify the expression

Simplify \((\csc^{2}x - 1)-2\cot x+1\) by combining like - terms.
\(-1 + 1\) cancels out, and we get \(\csc^{2}x-2\cot x\)

Answer:

\(\csc^{2}x-2\cot x\)