If $$\sin \alpha + \sin \beta = \cos \alpha + \cos \beta = 1$$, then $$\sin \alpha + \cos \alpha =$$?

$$\sin\alpha+\sin\beta=1$$

$$\sin^2\alpha+\sin^2\beta+2\sin\alpha\ \sin\beta\ =1$$......(1)

$$\cos\alpha+\cos\beta=1$$

$$\cos^2\alpha+\cos^2\beta+2\cos\alpha\ \cos\beta\ =1$$......(2)

Adding (1) and (2),

$$\left(\sin^2\alpha\ +\cos^2\alpha\right)+\left(\sin^2\beta\ +\cos^2\beta\right)+2\sin\alpha\ \sin\beta\ +2\cos\alpha\ \cos\beta\ =1+1$$

$$1+1+2\sin\alpha\ \sin\beta\ +2\cos\alpha\ \cos\beta\ =2$$

$$2\left[\cos\alpha\ \cos\beta+\sin\alpha\ \sin\beta\right]=0$$

$$\cos\left(\beta-\alpha\right)=0$$

$$\beta-\alpha=90^{\circ\ }$$

$$\beta\ =90^{\circ\ }+\alpha\ $$

$$\sin\alpha+\sin\beta=1$$

$$\sin\alpha+\sin\left(90^{\circ}-\alpha\ \right)=1$$

$$\sin\alpha+\cos\alpha=1$$

Hence, the correct answer is Option C