Simplify (1-sin(x)^2)/(sin(x)-csc(x))

Problem

(1−sin^2(x))/(sin(x)−csc(x))

Solution

1. Apply the Pythagorean identity to the numerator by substituting 1−sin^2(x)=cos^2(x)

2. Rewrite the cosecant function in the denominator using the reciprocal identity csc(x)=1/sin(x)

3. Find a common denominator for the terms in the denominator of the main fraction.

sin(x)−1/sin(x)=(sin^2(x)−1)/sin(x)

4. Apply the Pythagorean identity again to the denominator's numerator, noting that sin^2(x)−1=−cos^2(x)

(sin^2(x)−1)/sin(x)=(−cos^2(x))/sin(x)

5. Divide the fractions by multiplying the numerator by the reciprocal of the denominator.

cos^2(x)⋅sin(x)/(−cos^2(x))

6. Simplify the expression by canceling the common cos^2(x) terms.

Final Answer

(1−sin^2(x))/(sin(x)−csc(x))=−sin(x)

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