Verify the Identity

Problem

cot2(x)/(csc(x)−1)=(1+sin(x))/sin(x)

Solution

1. Use the Pythagorean identity to rewrite the numerator of the left side.

cot2(x)=csc2(x)−1

2. Substitute the identity into the expression.

(csc2(x)−1)/(csc(x)−1)

3. Factor the numerator as a difference of squares.

csc2(x)−1=(csc(x)−1)*(csc(x)+1)

4. Simplify the fraction by canceling the common factor in the numerator and denominator.

((csc(x)−1)*(csc(x)+1))/(csc(x)−1)=csc(x)+1

5. Rewrite the expression in terms of sine using the reciprocal identity.

csc(x)+1=1/sin(x)+1

6. Find a common denominator to combine the terms.

1/sin(x)+sin(x)/sin(x)=(1+sin(x))/sin(x)

Final Answer

cot2(x)/(csc(x)−1)=(1+sin(x))/sin(x)

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