Verify the Identity

Problem

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

Solution

1. Find a common denominator for the two fractions on the left side of the equation.

Common Denominator=sin(x)*(1+cos(x))

2. Rewrite the fractions using the common denominator by multiplying the first term by sin(x)/sin(x) and the second term by (1+cos(x))/(1+cos(x))

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

3. Expand the numerator by squaring the binomial term (1+cos(x))2

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

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

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

5. Combine like terms in the numerator.

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

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

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

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

2/sin(x)

8. Apply the reciprocal identity 1/sin(x)=csc(x) to reach the final form.

2*csc(x)

Final Answer

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

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