Simplify cot(x)^2

Problem

cot2(x)

Solution

1. Identify the expression as the square of the cotangent function, which is defined as the ratio of the cosine function to the sine function.

2. Apply the quotient identity for cotangent, which states that cot(x)=cos(x)/sin(x)

3. Square the identity to express the original term in terms of sine and cosine.

cot2(x)=cos2(x)/sin2(x)

4. Apply the Pythagorean identity cos2(x)=1−sin2(x) to rewrite the numerator.

cot2(x)=(1−sin2(x))/sin2(x)

5. Distribute the denominator to separate the fraction into two terms.

cot2(x)=1/sin2(x)−sin2(x)/sin2(x)

6. Simplify using the reciprocal identity csc2(x)=1/sin2(x) and the fact that any non-zero value divided by itself is 1.

cot2(x)=csc2(x)−1

Final Answer

cot2(x)=csc2(x)−1

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