Simplify (sec(x))/(cos(x))-(tan(x))/(cot(x))

Problem

sec(x)/cos(x)−tan(x)/cot(x)

Solution

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

2. Substitute these identities into the expression to create complex fractions.

1/cos(x)/cos(x)−tan(x)/1/tan(x)

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

1/cos2(x)−tan2(x)

4. Apply the identity 1/cos2(x)=sec2(x) to rewrite the first term.

sec2(x)−tan2(x)

5. Use the Pythagorean identity sec2(x)=1+tan2(x) to simplify the final expression.

1+tan2(x)−tan2(x)

6. Combine like terms to reach the final result.

1

Final Answer

sec(x)/cos(x)−tan(x)/cot(x)=1

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