Expand the Trigonometric Expression sec(x)^4-tan(x)^4

Problem

sec4(x)−tan4(x)

Solution

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

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

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

3. Apply the Pythagorean identity sec2(x)=1+tan2(x) which implies sec2(x)−tan2(x)=1

sec4(x)−tan4(x)=(1)*(sec2(x)+tan2(x))

4. Simplify the expression by removing the factor of 1.

sec4(x)−tan4(x)=sec2(x)+tan2(x)

5. Substitute sec2(x)=1+tan2(x) again if a single trigonometric function is desired.

sec4(x)−tan4(x)=(1+tan2(x))+tan2(x)

6. Combine like terms to reach the expanded form.

sec4(x)−tan4(x)=1+2*tan2(x)

Final Answer

sec4(x)−tan4(x)=1+2*tan2(x)

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