Factor tan(x)^4-sec(x)^4

Problem

tan4(x)−sec4(x)

Solution

1. Identify the expression as a difference of squares, which follows the pattern a2−b2=(a−b)*(a+b)

2. Rewrite the terms as squares of squares to prepare for factoring.

tan4(x)−sec4(x)=(tan2(x))2−(sec2(x))2

3. Apply the difference of squares formula using a=tan2(x) and b=sec2(x)

(tan2(x)−sec2(x))*(tan2(x)+sec2(x))

4. Simplify the first factor using the Pythagorean identity 1+tan2(x)=sec2(x) which implies tan2(x)−sec2(x)=−1

−1⋅(tan2(x)+sec2(x))

5. Distribute the negative sign to reach the final factored and simplified form.

−(tan2(x)+sec2(x))

Final Answer

tan4(x)−sec4(x)=−(tan2(x)+sec2(x))

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