Find $\lim_{{x \to 0}} \frac{x - \sin x}{x - \tan x}$ without using L'Hôpital's rule

I'm trying to find this limit:

$$\lim_{{x \to 0}} \frac{x - \sin x}{x - \tan x}$$

I want to use only algebraic manipulations and trigonometric identities, without L'Hôpital's rule or Taylor series.

I tried to use all the identities I know but I couldn't get rid of the indeterminate form.