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

Problem

(1+sin(u))/cos(u)+cos(u)/(1+sin(u))

Solution

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

((1+sin(u))2+cos2(u))/(cos(u)*(1+sin(u)))

2. Expand the numerator using the square of a binomial formula (a+b)2=a2+2*a*b+b2

(1+2*sin(u)+sin2(u)+cos2(u))/(cos(u)*(1+sin(u)))

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

(1+2*sin(u)+1)/(cos(u)*(1+sin(u)))

4. Combine like terms in the numerator.

(2+2*sin(u))/(cos(u)*(1+sin(u)))

5. Factor out the constant 2 from the numerator.

(2*(1+sin(u)))/(cos(u)*(1+sin(u)))

6. Cancel the common factor (1+sin(u)) from the numerator and denominator.

2/cos(u)

7. Use the reciprocal identity 1/cos(u)=sec(u) to write the final expression.

2*sec(u)

Final Answer

(1+sin(u))/cos(u)+cos(u)/(1+sin(u))=2*sec(u)

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