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

Problem

(cos(x)*cot(x))/(1−sin(x))−1=csc(x)

Solution

1. Substitute the cotangent identity cot(x)=cos(x)/sin(x) into the numerator of the fraction.

(cos(x)⋅cos(x)/sin(x))/(1−sin(x))−1

2. Simplify the numerator by multiplying the cosine terms to get cos2(x)

cos2(x)/sin(x)/(1−sin(x))−1

3. Apply the Pythagorean identity cos2(x)=1−sin2(x) to rewrite the numerator.

(1−sin2(x))/sin(x)/(1−sin(x))−1

4. Factor the difference of squares 1−sin2(x) as (1−sin(x))*(1+sin(x))

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

5. Cancel the common factor (1−sin(x)) from the numerator and the denominator.

(1+sin(x))/sin(x)−1

6. Split the fraction into two separate terms.

1/sin(x)+sin(x)/sin(x)−1

7. Simplify the expression by noting that sin(x)/sin(x)=1 and using the reciprocal identity 1/sin(x)=csc(x)

csc(x)+1−1

8. Combine like terms to reach the final simplified form.

csc(x)

Final Answer

(cos(x)*cot(x))/(1−sin(x))−1=csc(x)

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