2. Suppose $\theta$ is an angle between 0 and $\frac{\pi}{2}$, and $\sin 2 \theta=a$. Which of the following expressions is equal to $\sin \theta+\cos \theta$ ?
(A) $\sqrt{a+1}$
(B) $(\sqrt{2}-1) a+1$
(C) $\sqrt{a+1}-\sqrt{a^{2}-a}$
(D) $\sqrt{a+1}+\sqrt{a^{2}-a}$
(E) None of the above
2. Answer: (A)
Note that
$$
(\sin \theta+\cos \theta)^{2}=\sin ^{2} \theta+\cos ^{2} \theta+2 \sin \theta \cos \theta=1+\sin 2 \theta=1+a \text {. }
$$

Since $0 \leq \theta \leq \frac{\pi}{2}$, we have $\sin \theta+\cos \theta>0$. So $\sin \theta+\cos \theta=\sqrt{1+a}$.