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

Problem

(1−cos(x))/sin(x)=csc(x)−cot(x)

Solution

1. Split the fraction on the left side of the equation into two separate terms by dividing each term in the numerator by the denominator.

(1−cos(x))/sin(x)=1/sin(x)−cos(x)/sin(x)

2. Apply the reciprocal identity for the first term, where the reciprocal of the sine function is the cosecant function.

1/sin(x)=csc(x)

3. Apply the quotient identity for the second term, where the ratio of the cosine function to the sine function is the cotangent function.

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

4. Substitute these identities back into the expression to complete the verification.

csc(x)−cot(x)=csc(x)−cot(x)

Final Answer

(1−cos(x))/sin(x)=csc(x)−cot(x)

Want more problems? Check here!