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

Problem

(sec(x)−cos(x))/tan(x)

Solution

1. Rewrite the expression using basic trigonometric identities for sec(x) and tan(x)

sec(x)=1/cos(x)

tan(x)=sin(x)/cos(x)

2. Substitute these identities into the original expression.

(1/cos(x)−cos(x))/sin(x)/cos(x)

3. Simplify the numerator by finding a common denominator of cos(x)

1/cos(x)−cos(x)=(1−cos2(x))/cos(x)

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

sin2(x)/cos(x)

5. Divide the simplified numerator by the denominator sin(x)/cos(x) by multiplying by the reciprocal.

sin2(x)/cos(x)⋅cos(x)/sin(x)

6. Cancel the common factors of cos(x) and one factor of sin(x)

sin2(x)/sin(x)=sin(x)

Final Answer

(sec(x)−cos(x))/tan(x)=sin(x)

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