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

Problem

(1−tan^2(x))/(1+tan^2(x))

Solution

1. Identify the trigonometric identity for the denominator using the Pythagorean identity 1+tan^2(x)=sec^2(x)

(1−tan^2(x))/sec^2(x)

2. Rewrite the expression by splitting the fraction into two separate terms.

1/sec^2(x)−tan^2(x)/sec^2(x)

3. Convert all terms into sine and cosine using the identities 1/sec(x)=cos(x) and tan(x)=sin(x)/cos(x)

cos^2(x)−sin^2(x)/cos^2(x)/1/cos^2(x)

4. Simplify the second term by canceling the cos^2(x) denominators.

cos^2(x)−sin^2(x)

5. Apply the double angle identity for cosine, which states cos(2*x)=cos^2(x)−sin^2(x)

cos(2*x)

Final Answer

(1−tan^2(x))/(1+tan^2(x))=cos(2*x)

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