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

Problem

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

Solution

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

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

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

(cos(x)*(1−sin(x)))/(1−sin2(x))

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

(cos(x)*(1−sin(x)))/cos2(x)

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

(1−sin(x))/cos(x)

5. Split the fraction into two separate terms to simplify further.

1/cos(x)−sin(x)/cos(x)

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

sec(x)−tan(x)

Final Answer

cos(x)/(1+sin(x))=sec(x)−tan(x)

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