### Instruction: Among all triangles $ABC,$ find the maximum value of $\sin A + \sin B \sin C.$.### Response: To maximize $\sin A + \sin B \sin C,$ I want to make both terms as large as possible.
I know that $\sin A$ is maximized when $A$ is a right angle, so I will assume that $A = \frac{\pi}{2}.$
Then, I have $\sin A = 1$ and $\sin B \sin C = \cos B \cos C.$
I want to maximize $\cos B \cos C$ subject to the constraint that $B + C = \frac{\pi}{2}.$
I recall that $\cos x$ is a concave function on $[0, \frac{\pi}{2}],$ so by Jensen's inequality, I have $\cos B \cos C \leq \cos \left(\frac{B + C}{2}\right)^2.$
The equality holds when $B = C = \frac{\pi}{4}.$
Therefore, the maximum value of $\sin A + \sin B \sin C$ is $1 + \cos \left(\frac{\pi}{4}\right)^2 = 1 + \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{3}{2}.$