Simplify (tan(x)+sec(x))(tan(x)-sec(x))

Problem

(tan(x)+sec(x))*(tan(x)−sec(x))

Solution

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

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

(tan(x)+sec(x))*(tan(x)−sec(x))=tan^2(x)−sec^2(x)

3. Recall the fundamental Pythagorean identity relating tangent and secant.

1+tan^2(x)=sec^2(x)

4. Rearrange the identity to match the expression by subtracting sec^2(x) and 1 from both sides.

tan^2(x)−sec^2(x)=−1

Final Answer

(tan(x)+sec(x))*(tan(x)−sec(x))=−1

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