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

Problem

(1+tan2(x))/(1+cot2(x))

Solution

1. Apply Pythagorean identities to the numerator and denominator. Recall that 1+tan2(x)=sec2(x) and 1+cot2(x)=csc2(x)

(1+tan2(x))/(1+cot2(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)

sec2(x)/csc2(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=sin2(x)/cos2(x)

4. Apply the quotient identity for tangent, where sin(x)/cos(x)=tan(x)

sin2(x)/cos2(x)=tan2(x)

Final Answer

(1+tan2(x))/(1+cot2(x))=tan2(x)

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