Simplify (sin(x)-cos(x))(sin(x)+cos(x))

Problem

(sin(x)−cos(x))*(sin(x)+cos(x))

Solution

1. Identify the expression as a product of a sum and a difference, which follows the algebraic pattern (a−b)*(a+b)=a2−b2

2. Apply the formula for the difference of squares where a=sin(x) and b=cos(x)

3. Expand the expression using this identity.

(sin(x)−cos(x))*(sin(x)+cos(x))=sin2(x)−cos2(x)

4. Recognize that this result is a known form of the double-angle identity for cosine, specifically cos(2*x)=cos2(x)−sin2(x)

5. Factor out a negative sign to match the identity.

−(cos2(x)−sin2(x))=−cos(2*x)

Final Answer

(sin(x)−cos(x))*(sin(x)+cos(x))=−cos(2*x)

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