Simplify 1-tan(x)^2

Problem

1−tan2(x)

Solution

1. Identify the trigonometric identity for tan(x) in terms of sin(x) and cos(x)

tan(x)=sin(x)/cos(x)

2. Substitute the identity into the expression.

1−(sin(x)/cos(x))2

3. Distribute the exponent to the numerator and denominator.

1−sin2(x)/cos2(x)

4. Find a common denominator to combine the terms.

cos2(x)/cos2(x)−sin2(x)/cos2(x)

5. Combine the fractions into a single expression.

(cos2(x)−sin2(x))/cos2(x)

6. Apply the double-angle identity for cosine, where cos(2*x)=cos2(x)−sin2(x)

cos(2*x)/cos2(x)

7. Rewrite using the secant identity sec2(x)=1/cos2(x) if desired, or leave in terms of cosine.

cos(2*x)*sec2(x)

Final Answer

1−tan2(x)=cos(2*x)/cos2(x)

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