Simplify (sin(theta)-cos(theta))^2

Problem

(sin(θ)−cos(θ))2

Solution

1. Expand the square using the algebraic identity (a−b)2=a2−2*a*b+b2

sin2(θ)−2*sin(θ)*cos(θ)+cos2(θ)

2. Rearrange the terms to group the squared trigonometric functions together.

sin2(θ)+cos2(θ)−2*sin(θ)*cos(θ)

3. Apply the Pythagorean identity sin2(θ)+cos2(θ)=1

1−2*sin(θ)*cos(θ)

4. Apply the double-angle identity sin(2*θ)=2*sin(θ)*cos(θ) to further simplify the expression.

1−sin(2*θ)

Final Answer

(sin(θ)−cos(θ))2=1−sin(2*θ)

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