Simplify (sec(x)-1)(sec(x)+1)

Problem

(sec(x)−1)*(sec(x)+1)

Solution

1. Recognize the pattern as a difference of squares, which follows the algebraic identity (a−b)*(a+b)=a^2−b^2

2. Apply the identity by letting a=sec(x) and b=1

(sec(x)−1)*(sec(x)+1)=sec^2(x)−1^2

3. Simplify the constant by calculating 1^2=1

sec^2(x)−1

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

tan^2(x)

Final Answer

(sec(x)−1)*(sec(x)+1)=tan^2(x)

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