Simplify

Problem

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

Solution

1. Multiply the numerators and denominators of the two fractions.

(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)=a^2−b^2

(sin(x)*(1−cos(x)))/(1−cos^2(x))

3. Apply the Pythagorean identity sin^2(x)+cos^2(x)=1 which implies 1−cos^2(x)=sin^2(x)

(sin(x)*(1−cos(x)))/sin^2(x)

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

(1−cos(x))/sin(x)

5. Distribute the denominator to rewrite the expression as two separate terms.

1/sin(x)−cos(x)/sin(x)

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

csc(x)−cot(x)

Final Answer

sin(x)/(1+cos(x))⋅(1−cos(x))/(1−cos(x))=csc(x)−cot(x)

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