Given, tan x + cot x

= sin x/cos x + cos x/sinx

= (sin² x + cos² x)/(sin x * cos x)       {since sin² x + cos² x = 1}

= 1/(sin x * cos x)

= (2 * 1)/(2 *sin x * cos x)

= 2/sin 2x                                          {since sin 2x = 2 * sin x * cos x}

Let y = 2/sin 2x

=> sin 2x = 2/y

Now, -1 ≤ 2/y ≤ 1           {since -1 ≤ sin 2x ≤ 1}

=> -1/2 ≤ 1/y ≤ 1/2

=> -2 ≥ y ≥ 2

=> 2 ≤ y ≤ -2

=> y ∈ [-2, 2]

So, the range of tan x + cot x is [-2, 2]