Simplify (1+tan(x)^2)/(csc(x)^2)

Problem

(1+tan2(x))/csc2(x)

Solution

1. Apply the Pythagorean identity to the numerator using the identity 1+tan2(x)=sec2(x)

sec2(x)/csc2(x)

2. Rewrite in terms of sine and cosine using the reciprocal identities sec(x)=1/cos(x) and csc(x)=1/sin(x)

1/cos2(x)/1/sin2(x)

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

1/cos2(x)⋅sin2(x)/1

4. Combine the terms into a single fraction.

sin2(x)/cos2(x)

5. Apply the quotient identity sin(x)/cos(x)=tan(x) to reach the simplest form.

tan2(x)

Final Answer

(1+tan2(x))/csc2(x)=tan2(x)

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