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

Problem

1/(1+sin(x))−1/(1−sin(x))

Solution

1. Find a common denominator by multiplying the two denominators together.

D=(1+sin(x))*(1−sin(x))

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

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

3. Simplify the numerator by distributing the negative sign and combining like terms.

1−sin(x)−1−sin(x)=−2*sin(x)

4. Simplify the denominator using the difference of squares identity (a+b)*(a−b)=a2−b2

(1+sin(x))*(1−sin(x))=1−sin2(x)

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

(−2*sin(x))/cos2(x)

6. Rewrite the expression using trigonometric definitions tan(x)=sin(x)/cos(x) and sec(x)=1/cos(x)

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

Final Answer

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

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