Simplify 1/(sin(x))-sin(x)

Problem

1/sin(x)−sin(x)

Solution

1. Identify the common denominator needed to subtract the terms, which is sin(x)

2. Rewrite the second term as a fraction by multiplying and dividing by sin(x)

1/sin(x)−sin2(x)/sin(x)

3. Combine the fractions into a single expression over the common denominator.

(1−sin2(x))/sin(x)

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

cos2(x)/sin(x)

5. Decompose the fraction to express the result in terms of basic trigonometric functions.

cos(x)⋅cos(x)/sin(x)

6. Substitute the cotangent identity cos(x)/sin(x)=cot(x)

cos(x)*cot(x)

Final Answer

1/sin(x)−sin(x)=cos(x)*cot(x)

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