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

Problem

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

Solution

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

((1+cos(x))2+sin2(x))/(sin(x)*(1+cos(x)))

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

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

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

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

4. Combine like terms in the numerator.

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

5. Factor out the common factor of 2 from the numerator.

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

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

2/sin(x)

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

2*csc(x)

Final Answer

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

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