Simplify (1+cos(x))(1-cos(x))

Problem

(1+cos(x))*(1−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)=a^2−b^2

2. Apply the formula for the difference of squares where a=1 and b=cos(x)

1^2−cos^2(x)

3. Simplify the constant term 1^2 to 1

1−cos^2(x)

4. Apply the Pythagorean identity sin^2(x)+cos^2(x)=1 which implies that 1−cos^2(x)=sin^2(x)

sin^2(x)

Final Answer

(1+cos(x))*(1−cos(x))=sin^2(x)

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