1. The problem is to simplify the expression $\sin(x + \frac{\pi}{2})$. 2. We use the sine addition formula: $$\sin(a + b) = \sin a \cos b + \cos a \sin b$$ 3. Applying this formula with $a = x$ and $b = \frac{\pi}{2}$, we get: $$\sin\left(x + \frac{\pi}{2}\right) = \sin x \cos \frac{\pi}{2} + \cos x \sin \frac{\pi}{2}$$ 4. We know that $\cos \frac{\pi}{2} = 0$ and $\sin \frac{\pi}{2} = 1$, so: $$\sin x \cdot 0 + \cos x \cdot 1 = \cos x$$ 5. Therefore, the simplified expression is: $$\sin\left(x + \frac{\pi}{2}\right) = \cos x$$ 6. Among the given options, the correct answer is b. $\cos x$.