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

Problem

(1−tan2(x))/(1+tan2(x))+1

Solution

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

(1−tan2(x))/sec2(x)+1

2. Split the fraction by dividing each term in the numerator by the denominator.

1/sec2(x)−tan2(x)/sec2(x)+1

3. Use reciprocal and quotient identities to rewrite the terms in terms of sine and cosine, noting that 1/sec2(x)=cos2(x) and tan2(x)/sec2(x)=sin2(x)/cos2(x)⋅cos2(x)=sin2(x)

cos2(x)−sin2(x)+1

4. Apply the Pythagorean identity 1−sin2(x)=cos2(x) to simplify the expression further.

cos2(x)+cos2(x)

5. Combine like terms to reach the final simplified form.

2*cos2(x)

Final Answer

(1−tan2(x))/(1+tan2(x))+1=2*cos2(x)

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