How to prove that limit of tan x / x = 1 as x approaches 0 ?

Requirement

\[\lim _{x \rightarrow 0} \frac{\sin x}{x} = 1\]

Proof of limit of sin x / x = 1 as x approaches 0

Proof

By definition of tan x:

\[\frac{\tan x}{x} =\frac{\sin x}{x \cdot \cos x}=\frac{\sin x}{x} \times \frac{1}{\cos x} \\\] \[\begin{aligned} \lim_{x\to 0} \frac{\tan x}{x} &= \lim_{x\to 0} \left(\frac{\sin x}{x} \times \frac{1}{\cos x}\right)\\ &=1 \times 1\\ &=1 \end{aligned}\]

we conclude that:

\[\lim _{x \rightarrow 0} \frac{\tan x}{x} = 1\]