If `0 le x le pi`, where is `(tan x)/(sin x)` defined ?

Options

A. `0 le x le pi`
B. `0 lt x lt pi`
C. `0 lt x lt (pi)/(2)`
D. `0 lt x lt (pi)/(2)` and `(pi)/(2)lt x lt pi`

Answer

D

Explanation

Between 0 and `pi`, tangent is dedfined for anything but `(pi)/(2)`. Sine is defined for all values of x, but since the sine is in the denominator here, the expression `(tan x)/(sin x)` will be undefined when sin x = 0, which is when x = 0 or `pi`. So the expression `(tan x)/(sin x)` is defined when `0 lt x lt (pi)/(2)` or `(pi)/(2)lt x lt pi`.