Simplify csc(x)-1/(csc(x))

Problem

csc(x)−1/csc(x)

Solution

1. Identify the reciprocal identity for the second term, noting that 1/csc(x)=sin(x)

csc(x)−sin(x)

2. Rewrite the first term using the identity csc(x)=1/sin(x) to create a common denominator.

1/sin(x)−sin(x)

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

1/sin(x)−sin2(x)/sin(x)

4. Combine the fractions into a single expression.

(1−sin2(x))/sin(x)

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

cos2(x)/sin(x)

6. Decompose the fraction to express the result in terms of other trigonometric functions.

cos(x)⋅cos(x)/sin(x)

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

cos(x)*cot(x)

Final Answer

csc(x)−1/csc(x)=cos(x)*cot(x)

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