Factor sec(x)^4-2sec(x)^2tan(x)^2+tan(x)^4

Problem

sec(x)−2*sec(x)*tan(x)+tan(x)

Solution

1. Identify the expression as a perfect square trinomial of the form a^2−2*a*b+b^2

2. Define the terms where a=sec(x) and b=tan(x)

3. Apply the formula for a perfect square trinomial, which states a^2−2*a*b+b^2=(a−b)^2

4. Substitute the values back into the formula to get (sec(x)−tan(x))^2

5. Apply the Pythagorean identity sec(x)−tan(x)=1 to simplify the expression inside the parentheses.

6. Simplify the final result by calculating 1^2

Final Answer

sec(x)−2*sec(x)*tan(x)+tan(x)=1

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