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

Problem

(1+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)

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

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

(1+2*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.

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

4. Combine like terms in the numerator.

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

5. Factor out the constant 2 from the numerator.

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

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

2/cos(x)

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

2*sec(x)

Final Answer

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

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