Period of the Function:

To find the period of the given function, we need to determine the smallest positive value of x for which the function repeats its values.

Expressing the Function:

Let's start by expressing the given function:

f(x) = (cot(x/4) * tan(x/4)) / (1 - tan(x/2) - tan(x))

Simplifying the Function:

To simplify the function, we can use trigonometric identities:

1. cot(x) = 1 / tan(x)
2. tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b))

Applying these identities, we can simplify the function as follows:

f(x) = (cot(x/4) * tan(x/4)) / (1 - tan(x/2) - tan(x))
= ((1 / tan(x/4)) * tan(x/4)) / (1 - tan(x/2) - tan(x))
= 1 / (1 - tan(x/2) - tan(x))

Finding the Period:

To find the period of the function, we need to find the value of x for which the function repeats its values. In other words, we need to find the smallest positive value of x such that f(x) = f(x + T), where T represents the period of the function.

Let's substitute x = x + T into the function:

f(x + T) = 1 / (1 - tan((x + T)/2) - tan(x + T))

Since we want f(x + T) to be equal to f(x), we can set the two expressions equal to each other:

1 / (1 - tan(x/2) - tan(x)) = 1 / (1 - tan((x + T)/2) - tan(x + T))

Key Point:
The key point to notice here is that the denominator of the function is the same in both expressions.

Analyzing the Denominator:

Let's simplify the denominator of the function:

1 - tan(x/2) - tan(x) = 1 - tan(x/2) - [tan(x/2 + x/2)]
= 1 - tan(x/2) - [(tan(x/2) + tan(x/2)) / (1 - tan(x/2)tan(x/2))]
= 1 - tan(x/2) - 2tan(x/2) / (1 - tan(x/2)tan(x/2))
= (1 - 2tan(x/2)) / (1 - tan(x/2)tan(x/2))

Setting the Expressions Equal:

Now, let's set the two expressions equal to each other:

1 / (1 - tan(x/2) - tan(x)) = 1 / (1 - tan((x + T)/2) - tan(x + T))
=> (1 - 2tan(x/2)) / (1 - tan(x/2)tan(x/2)) = (1 - 2tan((x + T)/2)) / (1 - tan((x + T)/2)tan((x + T)/2))

Key Point:
The key point to notice here is that