Verify the Identity

Problem

cot2(t)/csc(t)=(1−sin2(t))/sin(t)

Solution

1. Rewrite the numerator on the left side using the quotient identity cot(t)=cos(t)/sin(t)

cot2(t)/csc(t)=cos2(t)/sin2(t)/csc(t)

2. Rewrite the denominator on the left side using the reciprocal identity csc(t)=1/sin(t)

cos2(t)/sin2(t)/1/sin(t)

3. Simplify the fraction by multiplying the numerator by the reciprocal of the denominator.

cos2(t)/sin2(t)⋅sin(t)/1

4. Cancel the common factor of sin(t) from the numerator and denominator.

cos2(t)/sin(t)

5. Apply the Pythagorean identity cos2(t)=1−sin2(t) to the numerator to match the right side of the identity.

(1−sin2(t))/sin(t)

Final Answer

cot2(t)/csc(t)=(1−sin2(t))/sin(t)

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