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

Problem

(sec(x)+tan(x))*(sec(x)−tan(x))

Solution

1. Identify the expression as a difference of squares in the form (a+b)*(a−b) where a=sec(x) and b=tan(x)

2. Apply the formula (a+b)*(a−b)=a^2−b^2 to expand the product.

sec^2(x)−tan^2(x)

3. Use the Pythagorean identity for trigonometric functions, which states that 1+tan^2(x)=sec^2(x)

4. Rearrange the identity to find the value of the expression by subtracting tan^2(x) from both sides.

sec^2(x)−tan^2(x)=1

Final Answer

(sec(x)+tan(x))*(sec(x)−tan(x))=1

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