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

Problem

sin(x)/(1+sin(x))−sin(x)/(1−sin(x))

Solution

1. Find a common denominator by multiplying the denominators of the two fractions.

Denominator=(1+sin(x))*(1−sin(x))

2. Rewrite the expression over the common denominator by cross-multiplying the numerators.

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

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

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

4. Simplify the numerator by combining like terms.

(sin(x)−sin2(x)−sin(x)−sin2(x))/(1−sin2(x))

(−2*sin2(x))/(1−sin2(x))

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

(−2*sin2(x))/cos2(x)

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

−2*tan2(x)

Final Answer

sin(x)/(1+sin(x))−sin(x)/(1−sin(x))=−2*tan2(x)

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