Simplify (1-cot(x))/(tan(x)-1)

Problem

(1−cot(x))/(tan(x)−1)

Solution

1. Express in terms of sine and cosine by using the quotient identities cot(x)=cos(x)/sin(x) and tan(x)=sin(x)/cos(x)

(1−cos(x)/sin(x))/(sin(x)/cos(x)−1)

2. Find common denominators for the numerator and the denominator of the complex fraction.

(sin(x)−cos(x))/sin(x)/(sin(x)−cos(x))/cos(x)

3. Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator.

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

4. Cancel the common factor (sin(x)−cos(x)) from the numerator and the denominator.

cos(x)/sin(x)

5. Apply the cotangent identity to write the final simplified expression.

cot(x)

Final Answer

(1−cot(x))/(tan(x)−1)=−cot(x)

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