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

Problem

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

Solution

1. Find a common denominator by multiplying the first fraction by (1+sin(x))/(1+sin(x)) and the second fraction by cos(x)/cos(x)

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

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

(sin(x)+sin2(x)+cos2(x))/(cos(x)*(1+sin(x)))

3. Apply the Pythagorean identity sin2(x)+cos2(x)=1 to simplify the numerator.

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

4. Cancel the common factor (1+sin(x)) from the numerator and the denominator.

1/cos(x)

5. Apply the reciprocal identity to express the result in terms of the secant function.

sec(x)

Final Answer

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

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