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

Problem

cot^2(x)−cos^2(x)=cot^2(x)*cos^2(x)

Solution

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

cot^2(x)−cos^2(x)=cos^2(x)/sin^2(x)−cos^2(x)

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

cos^2(x)*(1/sin^2(x)−1)

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

cos^2(x)*((1−sin^2(x))/sin^2(x))

4. Apply the Pythagorean identity sin^2(x)+cos^2(x)=1 which implies 1−sin^2(x)=cos^2(x) to simplify the numerator.

cos^2(x)*(cos^2(x)/sin^2(x))

5. Substitute the quotient identity cos^2(x)/sin^2(x)=cot^2(x) back into the expression to reach the final form.

cos^2(x)*cot^2(x)

Final Answer

cot^2(x)−cos^2(x)=cot^2(x)*cos^2(x)

Want more problems? Check here!