Simplify

Problem

(sin(θ)−cos(θ))−(sin(θ)+cos(θ))2

Solution

1. Expand the squared binomial term using the identity (a+b)2=a2+2*a*b+b2

(sin(θ)+cos(θ))2=sin2(θ)+2*sin(θ)*cos(θ)+cos2(θ)

2. Apply the Pythagorean identity sin2(θ)+cos2(θ)=1 to the expanded expression.

1+2*sin(θ)*cos(θ)

3. Substitute the simplified squared term back into the original expression.

(sin(θ)−cos(θ))−(1+2*sin(θ)*cos(θ))

4. Distribute the negative sign through the parentheses.

sin(θ)−cos(θ)−1−2*sin(θ)*cos(θ)

5. Rearrange the terms to present the final simplified form.

sin(θ)−cos(θ)−2*sin(θ)*cos(θ)−1

Final Answer

(sin(θ)−cos(θ))−(sin(θ)+cos(θ))2=sin(θ)−cos(θ)−2*sin(θ)*cos(θ)−1

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