Expand the Trigonometric Expression (sin(theta)-cos(theta))^2

Problem

(sin(θ)−cos(θ))^2

Solution

1. Apply the square of a binomial formula, which states (a−b)^2=a^2−2*a*b+b^2

2. Substitute a=sin(θ) and b=cos(θ) into the formula.

(sin(θ)−cos(θ))^2=sin^2(θ)−2*sin(θ)*cos(θ)+cos^2(θ)

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

(sin(θ)−cos(θ))^2=sin^2(θ)+cos^2(θ)−2*sin(θ)*cos(θ)

4. Apply the Pythagorean identity, where sin^2(θ)+cos^2(θ)=1

(sin(θ)−cos(θ))^2=1−2*sin(θ)*cos(θ)

5. Apply the double angle identity for sine, where 2*sin(θ)*cos(θ)=sin(2*θ)

(sin(θ)−cos(θ))^2=1−sin(2*θ)

Final Answer

(sin(θ)−cos(θ))^2=1−sin(2*θ)

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