Simplify 3/(sec(x)-tan(x))

Problem

3/(sec(x)−tan(x))

Solution

1. Identify the trigonometric identities for sec(x) and tan(x) to rewrite the expression in terms of sin(x) and cos(x)

sec(x)=1/cos(x)

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

2. Substitute these identities into the denominator of the original expression.

3/(1/cos(x)−sin(x)/cos(x))

3. Combine the fractions in the denominator since they share a common denominator of cos(x)

3/(1−sin(x))/cos(x)

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

(3*cos(x))/(1−sin(x))

5. Rationalize the denominator by multiplying both the numerator and the denominator by the conjugate 1+sin(x)

(3*cos(x)*(1+sin(x)))/((1−sin(x))*(1+sin(x)))

6. Expand the denominator using the difference of squares formula.

(3*cos(x)*(1+sin(x)))/(1−sin2(x))

7. Apply the Pythagorean identity 1−sin2(x)=cos2(x) to simplify the denominator.

(3*cos(x)*(1+sin(x)))/cos2(x)

8. Cancel the common factor of cos(x) from the numerator and the denominator.

(3*(1+sin(x)))/cos(x)

9. Distribute the division to separate the terms.

3/cos(x)+(3*sin(x))/cos(x)

10. Convert back to the basic trigonometric functions sec(x) and tan(x)

3*sec(x)+3*tan(x)

Final Answer

3/(sec(x)−tan(x))=3*(sec(x)+tan(x))

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