Simplify (1-tan(x)^4)/(1-tan(x)^2)

Problem

(1−tan^4(x))/(1−tan^2(x))

Solution

1. Identify the numerator as a difference of squares in the form a^2−b^2 where a=1 and b=tan^2(x)

2. Factor the numerator using the identity a^2−b^2=(a−b)*(a+b)

1−tan^4(x)=(1−tan^2(x))*(1+tan^2(x))

3. Substitute the factored form back into the original expression.

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

4. Cancel the common factor 1−tan^2(x) from the numerator and the denominator.

1+tan^2(x)

5. Apply the Pythagorean identity 1+tan^2(x)=sec^2(x) to simplify further.

sec^2(x)

Final Answer

(1−tan^4(x))/(1−tan^2(x))=sec^2(x)

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