Simplify (cos(x)-sec(x))/(sin(x))

Problem

(cos(x)−sec(x))/sin(x)

Solution

1. Express in terms of sine and cosine by using the reciprocal identity sec(x)=1/cos(x)

(cos(x)−1/cos(x))/sin(x)

2. Find a common denominator for the terms in the numerator to combine them into a single fraction.

(cos2(x)−1)/cos(x)/sin(x)

3. Apply the Pythagorean identity sin2(x)+cos2(x)=1 which implies cos2(x)−1=−sin2(x)

(−sin2(x))/cos(x)/sin(x)

4. Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator.

(−sin2(x))/(cos(x)*sin(x))

5. Cancel the common factor of sin(x) from the numerator and the denominator.

(−sin(x))/cos(x)

6. Apply the quotient identity tan(x)=sin(x)/cos(x) to reach the simplest form.

−tan(x)

Final Answer

(cos(x)−sec(x))/sin(x)=−tan(x)

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