Proving an algebra inequality with trig identities

[hearts123]

Hi everyone!
How can I prove that -sqrt(3) < (sqrt(3)*x + 1) / (sqrt(x^2 + 1)) <= 2 using trigonometric identities?
The question specifically stated that I had to use identities.
( <= is less than or equal to )
I was thinking that I could let x = something like cos α or sin α ??
Any help is appreciated!

 

[Dr.Peterson]

If no condition is given on x, you can't assume it's a sine or cosine of something. But there are other nice things to try.

The radicals make me think of the sine and cosine of pi/3 or pi/6 (your choice). So I'd divide everything by 2, to begin.

Then I'd divide each term in the middle by the denominator. Do x/sqrt(x^2+1) and 1/sqrt(x^2+1) sound like a sine and cosine, too?

Then play with it.