Solve (sinx+cosx)^2+(sinx-cosx)^2

$$( \sin x + \cos x ) ^ { 2 } + ( \sin x - \cos x ) ^ { 2 }$$

Factor

$$2$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sin(x)+\cos(x)\right)^{2}.

$$factor(\left(\sin(x)\right)^{2}+2\sin(x)\cos(x)+\left(\cos(x)\right)^{2}+\left(\sin(x)-\cos(x)\right)^{2})$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sin(x)-\cos(x)\right)^{2}.

$$factor(\left(\sin(x)\right)^{2}+2\sin(x)\cos(x)+\left(\cos(x)\right)^{2}+\left(\sin(x)\right)^{2}-2\sin(x)\cos(x)+\left(\cos(x)\right)^{2})$$

Combine \left(\sin(x)\right)^{2} and \left(\sin(x)\right)^{2} to get 2\left(\sin(x)\right)^{2}.

$$factor(2\left(\sin(x)\right)^{2}+2\sin(x)\cos(x)+\left(\cos(x)\right)^{2}-2\sin(x)\cos(x)+\left(\cos(x)\right)^{2})$$

Combine 2\sin(x)\cos(x) and -2\sin(x)\cos(x) to get 0.

$$factor(2\left(\sin(x)\right)^{2}+\left(\cos(x)\right)^{2}+\left(\cos(x)\right)^{2})$$

Combine \left(\cos(x)\right)^{2} and \left(\cos(x)\right)^{2} to get 2\left(\cos(x)\right)^{2}.

$$factor(2\left(\sin(x)\right)^{2}+2\left(\cos(x)\right)^{2})$$

Factor out 2.

$$2\left(\left(\sin(x)\right)^{2}+\left(\cos(x)\right)^{2}\right)$$

Evaluate

$$2$$