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

Problem

(csc(x)−cot(x))/(sec(x)−1)=cot(x)

Solution

1. Rewrite the terms on the left side using sine and cosine functions.

csc(x)=1/sin(x)

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

sec(x)=1/cos(x)

2. Substitute these expressions into the numerator and denominator of the left side.

(1/sin(x)−cos(x)/sin(x))/(1/cos(x)−1)

3. Simplify the numerator and denominator by finding common denominators.

(1−cos(x))/sin(x)/(1−cos(x))/cos(x)

4. Divide the fractions by multiplying the numerator by the reciprocal of the denominator.

(1−cos(x))/sin(x)⋅cos(x)/(1−cos(x))

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

cos(x)/sin(x)

6. Apply the cotangent identity to reach the final form.

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

Final Answer

(csc(x)−cot(x))/(sec(x)−1)=cot(x)

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