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

Problem

sin(x)/(1−cos(x))−sin(x)/(1+cos(x))

Solution

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

D=(1−cos(x))*(1+cos(x))

2. Rewrite the fractions over the common denominator by multiplying each numerator by the opposite denominator.

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

3. Distribute the sin(x) in the numerator.

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

4. Simplify the numerator by combining like terms.

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

(2*sin(x)*cos(x))/(1−cos2(x))

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

(2*sin(x)*cos(x))/sin2(x)

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

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

7. Apply the cotangent identity cos(x)/sin(x)=cot(x)

2*cot(x)

Final Answer

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

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