We simplify the given expression:

$\tan(x - \pi) \sin \left( x + \frac{\pi}{2} \right)$

Step 1: Use Trigonometric Identities

Identity for $\tan(x - \pi)$:

$\tan(x - \pi) = \tan x$ since $\tan$ has a period of $\pi$.

Thus, $\tan(x - \pi) = \tan x.$

Identity for $\sin(x + \pi/2)$:

$\sin(x + \pi/2) = \cos x$ (using the identity $\sin(x + 90^\circ) = \cos x$).

Step 2: Substitute the Values

$\tan(x - \pi) \sin(x + \pi/2) = \tan x \cos x.$

Step 3: Express in Terms of Sine and Cosine

Since $\tan x = \frac{\sin x}{\cos x}$, we rewrite: $\tan x \cos x = \frac{\sin x}{\cos x} \cdot \cos x.$

The $\cos x$ terms cancel, leaving: $\sin x.$

Final Answer:

$\sin x$

Would you like a deeper explanation or a graph?

Here are 5 related questions you might find interesting:

1. What is the identity for $\tan(a - b)$?
2. How do you prove $\sin(x + \pi) = -\sin x$?
3. What are the periodic properties of $\tan x$ and $\sin x$?
4. How do you simplify $\tan(x + \pi/4) \cos(x - \pi/4)$?
5. What is the derivative of $\tan x$?

Tip:

Remember that shifting trigonometric functions by $\pi$ or $\pi/2$ often simplifies expressions significantly!