Verify the Identity (csc(x))/(sin(x))-(cot(x))/(tan(x))=1

Problem

csc(x)/sin(x)−cot(x)/tan(x)=1

Solution

1. Rewrite the terms using reciprocal identities, where csc(x)=1/sin(x) and cot(x)=1/tan(x)

2. Substitute these identities into the left side of the equation.

1/sin(x)/sin(x)−1/tan(x)/tan(x)

3. Simplify the complex fractions by multiplying the numerators by the reciprocals of the denominators.

1/sin2(x)−1/tan2(x)

4. Apply the reciprocal identities 1/sin2(x)=csc2(x) and 1/tan2(x)=cot2(x)

csc2(x)−cot2(x)

5. Use the Pythagorean identity 1+cot2(x)=csc2(x) which can be rearranged to csc2(x)−cot2(x)=1

1=1

Final Answer

csc(x)/sin(x)−cot(x)/tan(x)=1

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