Introduction:

To find the maximum value of |tan(x) cot(x)|, we need to analyze the behavior of the functions involved and identify the critical points where the maximum occurs. Let's break down the problem step by step.

Step 1: Analyzing the functions:

• The function |tan(x)| represents the absolute value of the tangent function. It has a repeating pattern with vertical asymptotes at odd multiples of π/2 and horizontal asymptotes at even multiples of π.

• The function cot(x) represents the cotangent function. It has vertical asymptotes at multiples of π and no horizontal asymptotes.

Step 2: Combining the functions:

To find the maximum value of |tan(x) cot(x)|, we need to multiply the two functions together. Since the cotangent function has no horizontal asymptotes, we must consider the vertical asymptotes of the tangent function when determining the maximum.

Step 3: Identifying the critical points:

The critical points occur where either of the functions involved is undefined or equal to zero. In this case, we need to consider the vertical asymptotes of the tangent function.

• The vertical asymptotes of the tangent function occur at odd multiples of π/2: π/2, 3π/2, 5π/2, etc.

Step 4: Analyzing the behavior around critical points:

To determine the maximum value, we need to examine the behavior of the combined function around the critical points.

• At the vertical asymptotes of the tangent function, the value of |tan(x) cot(x)| approaches infinity from both sides. Therefore, there is no maximum value at these points.

Conclusion:

In summary, the function |tan(x) cot(x)| does not have a maximum value. It approaches infinity at the vertical asymptotes of the tangent function and has no horizontal asymptotes. Therefore, the maximum value is undefined.