Verify the Identity

Problem

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

Solution

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

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

2. Rewrite the fractions using the common denominator by multiplying the numerator and denominator of each term accordingly.

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

3. Distribute the sine function in the numerator.

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

4. Combine like terms in the numerator to simplify the expression.

(2*sin(x))/((1−cos(x))*(1+cos(x)))

5. Expand the denominator using the difference of squares formula.

(2*sin(x))/(1−cos2(x))

6. Apply the Pythagorean identity sin2(x)+cos2(x)=1 which implies 1−cos2(x)=sin2(x)

(2*sin(x))/sin2(x)

7. Simplify the fraction by canceling one sin(x) term from the numerator and denominator.

2/sin(x)

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

2*csc(x)

Final Answer

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

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