Verify the Identity csc(x)-cos(x)cot(x)=sin(x)

Problem

csc(x)−cos(x)*cot(x)=sin(x)

Solution

1. Express the trigonometric functions csc(x) and cot(x) in terms of sin(x) and cos(x)

csc(x)=1/sin(x)

cot(x)=cos(x)/sin(x)

2. Substitute these expressions into the left side of the identity.

1/sin(x)−cos(x)⋅cos(x)/sin(x)

3. Multiply the terms in the second part of the expression.

1/sin(x)−cos2(x)/sin(x)

4. Combine the fractions since they share a common denominator.

(1−cos2(x))/sin(x)

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

sin2(x)/sin(x)

6. Simplify the fraction by canceling one factor of sin(x)

sin(x)

Final Answer

csc(x)−cos(x)*cot(x)=sin(x)

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