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

Problem

tan^4(x)−sec^4(x)

Solution

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

2. Factor the expression using the difference of squares formula a^2−b^2=(a−b)*(a+b)

tan^4(x)−sec^4(x)=(tan^2(x)−sec^2(x))*(tan^2(x)+sec^2(x))

3. Apply the Pythagorean identity tan^2(x)+1=sec^2(x) which can be rearranged to tan^2(x)−sec^2(x)=−1

4. Substitute the value −1 into the factored expression.

−1⋅(tan^2(x)+sec^2(x))

5. Distribute the negative sign to simplify the final expression.

−tan^2(x)−sec^2(x)

Final Answer

tan^4(x)−sec^4(x)=−tan^2(x)−sec^2(x)

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