What is the range of the cosecant function?

The range of the cosecant function is indeed [1, ∞) in union with (-∞, -1]. This can be understood by first recalling that the cosecant function, which is the reciprocal of the sine function (csc(x) = 1/sin(x)), is defined wherever the sine function is not zero.

The sine function oscillates between -1 and 1, resulting in csc(x) having values outside of this interval. Specifically, when sine takes values in the interval (0, 1], the cosecant function will yield values from 1 to ∞. Conversely, when sine takes values in the interval [-1, 0), cosecant will give rise to values from -∞ to -1. Therefore, csc(x) can never equal values between -1 and 1, which means those numbers are not included in its range.

This leads to the conclusion that the correct range for the cosecant function is the union of values starting from 1 to positive infinity and from negative infinity up to -1.