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

Problem

sec2(x)/tan(x)=sec(x)*csc(x)

Solution

1. Express the left side of the identity in terms of sine and cosine using the definitions sec(x)=1/cos(x) and tan(x)=sin(x)/cos(x)

sec2(x)/tan(x)=1/cos2(x)/sin(x)/cos(x)

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

1/cos2(x)⋅cos(x)/sin(x)

3. Cancel the common factor of cos(x) from the numerator and the denominator.

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

4. Rewrite the expression as a product of two separate fractions to isolate the reciprocal identities.

1/cos(x)⋅1/sin(x)

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

sec(x)*csc(x)

Final Answer

sec2(x)/tan(x)=sec(x)*csc(x)

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