The problem is to solve the equation $\sin(x - \frac{\pi}{2}) = -\frac{\sqrt{2}}{2}$ for $x$.

1. Problem Description

The problem is to solve the equation $\sin(x - \frac{\pi}{2}) = -\frac{\sqrt{2}}{2}$ for $x$.

2. Solution Steps

We are given the equation $\sin(x - \frac{\pi}{2}) = -\frac{\sqrt{2}}{2}$.

First, we find the reference angle $\alpha$ such that $\sin(\alpha) = \frac{\sqrt{2}}{2}$.

We know that $\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$, so $\alpha = \frac{\pi}{4}$.

Since $\sin(x - \frac{\pi}{2})$ is negative, $x - \frac{\pi}{2}$ must be in the third or fourth quadrant. Therefore, we have two cases:

Case 1: $x - \frac{\pi}{2} = \pi + \frac{\pi}{4} + 2n\pi = \frac{5\pi}{4} + 2n\pi$, where $n$ is an integer.

Then, $x = \frac{5\pi}{4} + \frac{\pi}{2} + 2n\pi = \frac{5\pi}{4} + \frac{2\pi}{4} + 2n\pi = \frac{7\pi}{4} + 2n\pi$.

Case 2: $x - \frac{\pi}{2} = 2\pi - \frac{\pi}{4} + 2n\pi = \frac{7\pi}{4} + 2n\pi$, where $n$ is an integer.

Then, $x = \frac{7\pi}{4} + \frac{\pi}{2} + 2n\pi = \frac{7\pi}{4} + \frac{2\pi}{4} + 2n\pi = \frac{9\pi}{4} + 2n\pi = \frac{\pi}{4} + 2(n+1)\pi + 2\pi = \frac{9\pi}{4}$.

Since $x - \frac{\pi}{2}$ lies in the third or fourth quadrant, we need to consider the angles $\frac{5\pi}{4}$ and $\frac{7\pi}{4}$.

$x - \frac{\pi}{2} = \frac{5\pi}{4} + 2n\pi$ or $x - \frac{\pi}{2} = \frac{7\pi}{4} + 2n\pi$

$x = \frac{5\pi}{4} + \frac{\pi}{2} + 2n\pi = \frac{7\pi}{4} + 2n\pi$ or $x = \frac{7\pi}{4} + \frac{\pi}{2} + 2n\pi = \frac{9\pi}{4} + 2n\pi = \frac{\pi}{4} + 2(n+1)\pi$

When $n = 0$, $x = \frac{7\pi}{4}$ or $x = \frac{9\pi}{4}$.

The general solutions are $x = \frac{7\pi}{4} + 2n\pi$ and $x = \frac{9\pi}{4} + 2n\pi$.

Since $\sin(x-\pi/2) = -\cos(x)$, the equation becomes $-\cos(x) = -\frac{\sqrt{2}}{2}$ which means $\cos(x) = \frac{\sqrt{2}}{2}$. The solutions for this equation are $x = \frac{\pi}{4} + 2n\pi$ and $x = -\frac{\pi}{4} + 2n\pi = \frac{7\pi}{4} + 2(n-1)\pi$ for integers $n$.

Alternatively $\cos(x)=\frac{\sqrt{2}}{2}$, thus $x=\frac{\pi}{4}+2n\pi$ or $x=-\frac{\pi}{4}+2n\pi$

Then $x=\frac{\pi}{4}+2n\pi$ or $x=\frac{7\pi}{4}+2n\pi$ where $n$ is an integer

3. Final Answer

$x = \frac{7\pi}{4} + 2n\pi$ and $x = \frac{\pi}{4} + 2n\pi$