Factor 1/(sin(x))-sin(x)

Problem

1/sin(x)−sin(x)

Solution

1. Find a common denominator by treating the second term as a fraction over 1

1/sin(x)−(sin(x)⋅sin(x))/sin(x)

2. Combine the numerators over the shared denominator sin(x)

(1−sin2(x))/sin(x)

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

cos2(x)/sin(x)

4. Rewrite the expression as a product of two trigonometric ratios to show the factored form.

cos(x)⋅cos(x)/sin(x)

5. Substitute the identity for the cotangent function cot(x)=cos(x)/sin(x)

cos(x)*cot(x)

Final Answer

1/sin(x)−sin(x)=cos(x)*cot(x)

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