Verify the Identity (sec(x)-1)(sec(x)+1)=tan(x)^2

Problem

(sec(x)−1)*(sec(x)+1)=tan^2(x)

Solution

1. Identify the left side of the equation as a difference of squares in the form (a−b)*(a+b)

2. Expand the product using the difference of squares formula a^2−b^2

(sec(x)−1)*(sec(x)+1)=sec^2(x)−1^2

3. Simplify the constant term.

sec^2(x)−1^2=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)

sec^2(x)−1=tan^2(x)

Final Answer

(sec(x)−1)*(sec(x)+1)=tan^2(x)

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