What is the range of the function?

Calculation:

Given,

The function is\(y = \left( 1 - \cos(x) \right)^{-1} \), where \(x \neq 2n\pi \)and n  is an integer.

The cosine function has a range of  [-1, 1] , so cos(x) can take values from ( -1) to ( 1 ).

The expression \(1 - \cos(x) \) will take values from:

\( 1 - 1 = 0 \quad \text{to} \quad 1 - (-1) = 2 \)

So, (1 - cos(x)) takes values in the range \((0, 2] \), but \(x \neq 2n\pi \) excludes the value 0.

Since \(y = \frac{1}{1 - \cos(x)} \), the reciprocal function will take values in the range \(\left[ \frac{1}{2}, \infty \right) \)

∴ The range of the function is \([0.5, \infty) \).

The correct answer is Option (b)