Verify the Identity cot(x)^2-cos(x)^2=cot(x)^2cos(x)^2

Problem

cot2(x)−cos2(x)=cot2(x)*cos2(x)

Solution

1. Express the cotangent function in terms of sine and cosine using the quotient identity cot(x)=cos(x)/sin(x)

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

2. Factor out the common term cos2(x) from both terms in the expression.

cos2(x)*(1/sin2(x)−1)

3. Find a common denominator for the terms inside the parentheses to combine them into a single fraction.

cos2(x)*((1−sin2(x))/sin2(x))

4. Apply the Pythagorean identity sin2(x)+cos2(x)=1 which implies 1−sin2(x)=cos2(x) to simplify the numerator.

cos2(x)*(cos2(x)/sin2(x))

5. Substitute the quotient identity cos2(x)/sin2(x)=cot2(x) back into the expression to reach the final form.

cos2(x)*cot2(x)

Final Answer

cot2(x)−cos2(x)=cot2(x)*cos2(x)

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