The expression you provided is:

$\frac{1}{\tan}\cdot\left(5\cdot x-\pi\right)$

To interpret this expression correctly, it's important to determine how the components are intended to be grouped. Based on standard mathematical notation, let's clarify what the expression likely means.

Interpreting the Expression

If we consider the expression:

$\frac{1}{\tan}\cdot\left(5\cdot x-\pi\right)$

It can be interpreted as:

$\frac{1}{\tan(5x - \pi)}$

This is because the notation $\frac{1}{\tan}$ suggests taking the reciprocal of the tangent function applied to the entire expression $5x - \pi$.

Simplifying the Expression

To simplify $\frac{1}{\tan(5x - \pi)}$, we use the trigonometric identity that relates the tangent function and the cotangent function:

$\frac{1}{\tan(\theta)} = \cot(\theta)$

Thus, the expression can be rewritten as:

$\cot(5x - \pi)$

Conclusion

The expression $\frac{1}{\tan}(5x - \pi)$ simplifies to $\cot(5x - \pi)$. The cotangent function, $\cot(\theta)$, is the reciprocal of the tangent function, meaning it gives the ratio of the adjacent side to the opposite side in a right triangle or, equivalently, $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$.

If you have any more questions or would like additional details, please let me know!

Follow-Up Questions

1. Would you like to know how to graph the cotangent function?
2. Do you want to see an example of evaluating $\cot(5x - \pi)$ for a specific value of $x$?
3. Are you interested in understanding the periodicity of the cotangent function?
4. Would you like an explanation of how to convert between cotangent and other trigonometric functions?
5. Do you want to explore the derivatives or integrals involving the cotangent function?

Tip

Remember that understanding the reciprocal relationships between trigonometric functions, such as $\tan(\theta)$ and $\cot(\theta)$, can help simplify complex trigonometric expressions!