Verify the Identity

Problem

cot2(x)/(csc(x)+1)=(1−sin(x))/sin(x)

Solution

1. Apply the Pythagorean identity to the numerator of the left side by substituting cot2(x)=csc2(x)−1

(csc2(x)−1)/(csc(x)+1)

2. Factor the numerator as a difference of squares, where csc2(x)−1=(csc(x)−1)*(csc(x)+1)

((csc(x)−1)*(csc(x)+1))/(csc(x)+1)

3. Simplify the fraction by canceling the common factor (csc(x)+1) from the numerator and the denominator.

csc(x)−1

4. Rewrite in terms of sine by using the reciprocal identity csc(x)=1/sin(x)

1/sin(x)−1

5. Find a common denominator to combine the terms into a single fraction.

1/sin(x)−sin(x)/sin(x)

6. Subtract the numerators to reach the final form of the identity.

(1−sin(x))/sin(x)

Final Answer

cot2(x)/(csc(x)+1)=(1−sin(x))/sin(x)

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