Simplify (1-sin(x))(1+sin(x))

Problem

(1−sin(x))*(1+sin(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=sin(x)

1^2−sin^2(x)

3. Simplify the squares.

1−sin^2(x)

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

cos^2(x)

Final Answer

(1−sin(x))*(1+sin(x))=cos^2(x)

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