Simplify (cot(x))/(csc(x)-sin(x))

Problem

cot(x)/(csc(x)−sin(x))

Solution

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

cos(x)/sin(x)/(1/sin(x)−sin(x))

2. Find a common denominator for the expression in the denominator of the main fraction.

cos(x)/sin(x)/(1−sin2(x))/sin(x)

3. Apply the Pythagorean identity 1−sin2(x)=cos2(x) to simplify the denominator.

cos(x)/sin(x)/cos2(x)/sin(x)

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

cos(x)/sin(x)⋅sin(x)/cos2(x)

5. Cancel the common factors sin(x) and cos(x) from the numerator and denominator.

1/cos(x)

6. Identify the reciprocal identity 1/cos(x)=sec(x)

sec(x)

Final Answer

cot(x)/(csc(x)−sin(x))=sec(x)

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