Answer
$tan(x - \frac{\pi}{2}) = -cot ~x$
$sec(x - \frac{\pi}{2}) = csc ~x$
Work Step by Step
We can use figure 5 to see why the following formulas are true.
The graph of $tan(x - \frac{\pi}{2})$ is the graph of $tan~x$ shifted horizontally to the right by $\frac{\pi}{2}$
Then the vertical asymptotes are located at $x = \pi~n,$ where $n$ is an integer.
The points where $tan(x - \frac{\pi}{2}) = 0$ have the form $x = \frac{\pi}{2}+\pi n$
We can see that this closely matches up with the graph of $cot~x$ except that we need to flip the graph of $cot~x$ about the x axis in order to match up perfectly.
Therefore,
$tan(x - \frac{\pi}{2}) = -cot ~x$
The graph of $sec(x - \frac{\pi}{2})$ is the graph of $sec~x$ shifted horizontally to the right by $\frac{\pi}{2}$
Then the vertical asymptotes are located at $x = \pi~n,$ where $n$ is an integer.
The points where $sec(x - \frac{\pi}{2}) = 1$ have the form $x = \frac{\pi}{2}+\pi n$
We can see that the graph of $sec(x - \frac{\pi}{2})$ matches up perfectly with the graph of $csc~x$
Therefore,
$sec(x - \frac{\pi}{2}) = csc ~x$
