Simplify (cos(x))/(1-sin(x))-(cos(x))/(1+sin(x))

Problem

cos(x)/(1−sin(x))−cos(x)/(1+sin(x))

Solution

1. Find a common denominator by multiplying the two denominators, which results in a difference of squares.

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

2. Distribute the cos(x) in the numerator.

(cos(x)+cos(x)*sin(x)−(cos(x)−cos(x)*sin(x)))/(1−sin2(x))

3. Simplify the numerator by combining like terms.

(cos(x)+cos(x)*sin(x)−cos(x)+cos(x)*sin(x))/(1−sin2(x))

(2*cos(x)*sin(x))/(1−sin2(x))

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

(2*cos(x)*sin(x))/cos2(x)

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

(2*sin(x))/cos(x)

6. Use the tangent identity sin(x)/cos(x)=tan(x) to reach the final form.

2*tan(x)

Final Answer

cos(x)/(1−sin(x))−cos(x)/(1+sin(x))=2*tan(x)

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