Simplify (tan(x))/(1+sec(x))+(1+sec(x))/(tan(x))

Problem

tan(x)/(1+sec(x))+(1+sec(x))/tan(x)

Solution

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

(tan2(x)+(1+sec(x))2)/(tan(x)*(1+sec(x)))

2. Expand the numerator by squaring the binomial (1+sec(x))2

(tan2(x)+1+2*sec(x)+sec2(x))/(tan(x)*(1+sec(x)))

3. Apply the Pythagorean identity tan2(x)+1=sec2(x) to simplify the numerator.

(sec2(x)+2*sec(x)+sec2(x))/(tan(x)*(1+sec(x)))

4. Combine like terms in the numerator.

(2*sec2(x)+2*sec(x))/(tan(x)*(1+sec(x)))

5. Factor out the greatest common factor 2*sec(x) from the numerator.

(2*sec(x)*(sec(x)+1))/(tan(x)*(1+sec(x)))

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

(2*sec(x))/tan(x)

7. Rewrite in terms of sine and cosine using the definitions sec(x)=1/cos(x) and tan(x)=sin(x)/cos(x)

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

8. Simplify the complex fraction by multiplying by the reciprocal of the denominator.

2/cos(x)⋅cos(x)/sin(x)

9. Cancel the cosine terms and use the identity 1/sin(x)=csc(x)

2*csc(x)

Final Answer

tan(x)/(1+sec(x))+(1+sec(x))/tan(x)=2*csc(x)

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