Solve left(cos(x)+sin(x)right)^2+left(cos(x)-sin(x)right)^2

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

Factor

$$2$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

Factor out 2.

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

Evaluate

$$2$$