Simplify (1-sin(theta))(1+sin(theta))

Problem

(1−sin(θ))*(1+sin(θ))

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(θ)

1^2−sin^2(θ)

3. Simplify the constant term.

1−sin^2(θ)

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

cos^2(θ)

Final Answer

(1−sin(θ))*(1+sin(θ))=cos^2(θ)

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