Verify the Identity (csc(x)^2)/(cot(x))=csc(x)sec(x)

Problem

csc2(x)/cot(x)=csc(x)*sec(x)

Solution

1. Express the left side of the identity using the fundamental trigonometric definitions for cosecant and cotangent.

csc(x)=1/sin(x)

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

2. Substitute these definitions into the numerator and denominator of the expression.

csc2(x)/cot(x)=1/sin2(x)/cos(x)/sin(x)

3. Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator.

1/sin2(x)⋅sin(x)/cos(x)

4. Cancel the common factor of sin(x) from the numerator and the denominator.

1/(sin(x)*cos(x))

5. Rewrite the product as two separate fractions to identify the reciprocal identities.

1/sin(x)⋅1/cos(x)

6. Apply the reciprocal identities 1/sin(x)=csc(x) and 1/cos(x)=sec(x) to reach the final form.

csc(x)*sec(x)

Final Answer

csc2(x)/cot(x)=csc(x)*sec(x)

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