Simplify square root of (tan(x)^2+1)/(cot(x)^2+1)

Problem

√(,(tan2(x)+1)/(cot2(x)+1))

Solution

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

√(,sec2(x)/csc2(x))

2. Rewrite in terms of sine and cosine using the definitions 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.

√(,sin2(x)/cos2(x))

4. Apply the tangent identity sin(x)/cos(x)=tan(x) to simplify the expression inside the radical.

√(,tan2(x))

5. Take the square root of the squared term, noting that √(,u2)=|u|

|tan(x)|

Final Answer

√(,(tan2(x)+1)/(cot2(x)+1))=|tan(x)|

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