Verify the Identity (sin(x))/(1-cos(x))=csc(x)+cot(x)

Problem

sin(x)/(1−cos(x))=csc(x)+cot(x)

Solution

1. Multiply the numerator and the denominator of the left side by the conjugate of the denominator, which is 1+cos(x)

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

2. Expand the denominator using the difference of squares formula (a−b)*(a+b)=a2−b2

(sin(x)*(1+cos(x)))/(1−cos2(x))

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

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

4. Simplify the fraction by canceling one factor of sin(x) from the numerator and the denominator.

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

5. Split the fraction into two separate terms.

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

6. Substitute the reciprocal identity csc(x)=1/sin(x) and the quotient identity cot(x)=cos(x)/sin(x)

csc(x)+cot(x)

Final Answer

sin(x)/(1−cos(x))=csc(x)+cot(x)

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