see the details below.
Work Step by Step
Let $u=( \sin \theta, \cos \theta)$ and $v=(\cos \theta,-\sin \theta,)$, then we have since $$u\cdot v=\sin \theta \cos \theta-\cos \theta\sin \theta=0$$ then the set is orthogonal. Also, we have $$\|u\|=\sin^2\theta+\cos^2\theta=1,$$ $$\|u\|=\cos^2\theta+\sin^2\theta=1,$$ then the set is orthonormal. since the set is orthogonal and has two vectors and $R^2$ has dimension $2$ a then it is a basis for $R^2$.