Verify the Identity

Problem

sec(x)/sin(x)−sin(x)/cos(x)=cot(x)

Solution

1. Rewrite the secant function in terms of cosine using the reciprocal identity sec(x)=1/cos(x)

1/cos(x)/sin(x)−sin(x)/cos(x)=cot(x)

2. Simplify the first fraction by moving the cosine term to the denominator.

1/(sin(x)*cos(x))−sin(x)/cos(x)=cot(x)

3. Find a common denominator for the two fractions on the left side, which is sin(x)*cos(x)

1/(sin(x)*cos(x))−sin2(x)/(sin(x)*cos(x))=cot(x)

4. Combine the fractions over the common denominator.

(1−sin2(x))/(sin(x)*cos(x))=cot(x)

5. Apply the Pythagorean identity 1−sin2(x)=cos2(x) to the numerator.

cos2(x)/(sin(x)*cos(x))=cot(x)

6. Cancel a factor of cos(x) from the numerator and the denominator.

cos(x)/sin(x)=cot(x)

7. Apply the quotient identity cos(x)/sin(x)=cot(x) to complete the verification.

cot(x)=cot(x)

Final Answer

sec(x)/sin(x)−sin(x)/cos(x)=cot(x)

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