Simplify 1/(1+sin(x))

Problem

1/(1+sin(x))

Solution

1. Multiply the numerator and the denominator by the conjugate of the denominator, which is 1−sin(x)

1/(1+sin(x))⋅(1−sin(x))/(1−sin(x))

2. Expand the denominator using the difference of squares formula, (a+b)*(a−b)=a2−b2

(1−sin(x))/(1−sin2(x))

3. Apply the Pythagorean identity cos2(x)+sin2(x)=1 which implies 1−sin2(x)=cos2(x)

(1−sin(x))/cos2(x)

4. Split the fraction into two separate terms.

1/cos2(x)−sin(x)/cos2(x)

5. Rewrite the terms using reciprocal and quotient identities, where 1/cos2(x)=sec2(x) and sin(x)/cos2(x)=sin(x)/cos(x)⋅1/cos(x)=tan(x)*sec(x)

sec2(x)−sec(x)*tan(x)

Final Answer

1/(1+sin(x))=sec2(x)−sec(x)*tan(x)

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