Simplify csc(theta)-sin(theta)

Problem

csc(θ)−sin(θ)

Solution

1. Use the reciprocal identity to rewrite the cosecant function in terms of sine.

csc(θ)=1/sin(θ)

2. Substitute the identity back into the original expression.

1/sin(θ)−sin(θ)

3. Find a common denominator to combine the terms into a single fraction.

1/sin(θ)−sin2(θ)/sin(θ)

4. Combine the numerators over the common denominator.

(1−sin2(θ))/sin(θ)

5. Apply the Pythagorean identity where 1−sin2(θ)=cos2(θ)

cos2(θ)/sin(θ)

6. Rewrite the expression as a product of two fractions to identify further identities.

cos(θ)/sin(θ)⋅cos(θ)

7. Apply the cotangent identity where cos(θ)/sin(θ)=cot(θ)

cot(θ)*cos(θ)

Final Answer

csc(θ)−sin(θ)=cot(θ)*cos(θ)

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