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

Problem

tan4(x)−sec4(x)

Solution

1. Recognize the expression as a difference of squares in the form a2−b2 where a=tan2(x) and b=sec2(x)

2. Factor the expression using the difference of squares formula a2−b2=(a−b)*(a+b)

tan4(x)−sec4(x)=(tan2(x)−sec2(x))*(tan2(x)+sec2(x))

3. Apply the Pythagorean identity tan2(x)+1=sec2(x) which can be rearranged to tan2(x)−sec2(x)=−1

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

−1⋅(tan2(x)+sec2(x))

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

−tan2(x)−sec2(x)

Final Answer

tan4(x)−sec4(x)=−tan2(x)−sec2(x)

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