Simplify (1-cos(x))(csc(x)+cot(x))

Problem

(1−cos(x))*(csc(x)+cot(x))

Solution

1. Rewrite the trigonometric functions csc(x) and cot(x) in terms of sin(x) and cos(x)

csc(x)=1/sin(x)

cot(x)=cos(x)/sin(x)

2. Substitute these expressions back into the second factor of the original expression.

(1−cos(x))*(1/sin(x)+cos(x)/sin(x))

3. Combine the fractions inside the parentheses since they have a common denominator.

(1−cos(x))*((1+cos(x))/sin(x))

4. Multiply the numerators together.

((1−cos(x))*(1+cos(x)))/sin(x)

5. Apply the difference of squares formula (a−b)*(a+b)=a2−b2 to the numerator.

(1−cos2(x))/sin(x)

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

sin2(x)/sin(x)

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

sin(x)

Final Answer

(1−cos(x))*(csc(x)+cot(x))=sin(x)

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