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

Problem

(1−tan^2(x))/(1+tan^2(x))+1

Solution

1. Apply the Pythagorean identity to the denominator using the relation 1+tan^2(x)=sec^2(x)

(1−tan^2(x))/sec^2(x)+1

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

1/sec^2(x)−tan^2(x)/sec^2(x)+1

3. Use reciprocal and quotient identities to rewrite the terms in terms of sine and cosine, noting that 1/sec^2(x)=cos^2(x) and tan^2(x)/sec^2(x)=sin^2(x)/cos^2(x)⋅cos^2(x)=sin^2(x)

cos^2(x)−sin^2(x)+1

4. Apply the Pythagorean identity 1−sin^2(x)=cos^2(x) to simplify the expression further.

cos^2(x)+cos^2(x)

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

2*cos^2(x)

Final Answer

(1−tan^2(x))/(1+tan^2(x))+1=2*cos^2(x)

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