Simplify tan(x)-(sec(x)^2)/(tan(x))

Problem

tan(x)−sec2(x)/tan(x)

Solution

1. Find a common denominator by multiplying the first term by tan(x)/tan(x)

tan2(x)/tan(x)−sec2(x)/tan(x)

2. Combine the fractions into a single expression over the common denominator.

(tan2(x)−sec2(x))/tan(x)

3. Apply the Pythagorean identity tan2(x)+1=sec2(x) which can be rearranged to tan2(x)−sec2(x)=−1

(−1)/tan(x)

4. Use the reciprocal identity cot(x)=1/tan(x) to simplify the expression further.

−cot(x)

Final Answer

tan(x)−sec2(x)/tan(x)=−cot(x)

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