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

Problem

(1−cos(x))*(1+cos(x))

Solution

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

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

1^2−cos^2(x)

3. Simplify the squared term.

1−cos^2(x)

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