Simplify csc(x)-sin(x)

Problem

csc(x)−sin(x)

Solution

1. Rewrite the cosecant function in terms of sine using the reciprocal identity csc(x)=1/sin(x)

1/sin(x)−sin(x)

2. Find a common denominator by multiplying the second term by sin(x)/sin(x)

1/sin(x)−sin2(x)/sin(x)

3. Combine the fractions into a single expression.

(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 identify further trigonometric identities.

cos(x)/sin(x)⋅cos(x)

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

cot(x)*cos(x)

Final Answer

csc(x)−sin(x)=cot(x)*cos(x)

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