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The problem asks to determine the maximum value of \( P \), where \( P \) is given by:

\[
P = \frac{\sin x}{\cos x}
\]

This expression simplifies to:

\[
P = \tan x
\]

The function \( \tan x \) does not have a maximum value in the usual sense because \( \tan x \) can increase without bound as \( x \) approaches \( \frac{\pi}{2} \) from the left or decreases without bound as \( x \) approaches \( -\frac{\pi}{2} \) from the right within its period.

However, if the problem intends for \( x \) to be within a restricted domain, such as \( 0 \leq x \leq \frac{\pi}{2} \), then \( \tan x \) would increase from 0 to \( +\infty \) as \( x \) approaches \( \frac{\pi}{2} \).

Given no further domain restrictions, \( P = \tan x \) technically has no maximum value within its full domain because it can become arbitrarily large.

Would you like further details or have any specific questions?

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Here are 8 related questions to further explore the topic:

1. What is the period of the tangent function?
2. How does the tangent function behave as it approaches its asymptotes?
3. What are the general solutions to the equation \( \tan x = 1 \)?
4. How would you find the maximum of \( P = \tan x \) within a restricted interval, such as \( [0, \pi/4] \)?
5. What is the derivative of \( \tan x \), and how does it help understand the function's behavior?
6. Can \( \tan x \) have a local maximum or minimum within one period?
7. What are the key differences between \( \sin x \), \( \cos x \), and \( \tan x \) in terms of their ranges and periods?
8. How would you graph \( P = \tan x \) over one period and analyze its properties?

**Tip:** Remember that trigonometric functions often have periodic behavior, and understanding their intervals of increase and decrease is key to analyzing their maximum and minimum values.

Explore how to determine the maximum value of \( P = \tan x \) and understand its behavior within its domain. This problem delves into trigonometric functions and their periodic properties. Suitable for advanced high school students.

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