Simplify (sin(x)-cos(x))^2+(sin(x)+cos(x))^2

Problem

(sin(x)−cos(x))2+(sin(x)+cos(x))2

Solution

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

(sin(x)−cos(x))2=sin2(x)−2*sin(x)*cos(x)+cos2(x)

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

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

3. Combine the two expanded expressions by adding them together.

sin2(x)−2*sin(x)*cos(x)+cos2(x)+sin2(x)+2*sin(x)*cos(x)+cos2(x)

4. Cancel the middle terms −2*sin(x)*cos(x) and +2*sin(x)*cos(x) and group the remaining terms.

2*sin2(x)+2*cos2(x)

5. Factor out the common constant.

2*(sin2(x)+cos2(x))

6. Apply the Pythagorean identity sin2(x)+cos2(x)=1

2*(1)=2

Final Answer

(sin(x)−cos(x))2+(sin(x)+cos(x))2=2

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