You need to remember that `tan x = sin x/cos x ` such that:
`|sin x| =lt |x| =lt |sin x|/|cos x|`
Dividing by |sin x| yields: `1 =lt |x|/|sin x| =lt 1/|cos x|`
Evaluating the limit of terms of inequality yields:
`lim_(x-gt0) 1 =ltlim_(x-gt0)|x|/|sin x| =lt lim_(x-gt0) 1/|cos x|`
`` `1=ltlim_(x-gt0)|x|/|sin x| =lt 1/(cos 0) = 1/1 = 1`
You need to use squeeze principle to confirm that limit of `|x|/|sin x|` if x->0 is 1.
Using the squeeze pronciple it follows that `lim_(x-gt0)` `|x|/|sin x| = 1` , hence the inequality holds.