limit involving composition of Trigonometric function

Question

Evaluation of limit $\displaystyle \lim_{x\rightarrow 0}\frac{\cos (\tan x)-\cos x}{x^4}\;,$

without using L, Hospital Rule and series expansion.

I have solved it using $\displaystyle \cos (\tan x)-\cos x = -2\sin\left(\frac{\tan x+x}{2}\right)\cdot \sin \left(\frac{\tan x-x}{2}\right)$

Now Using $\displaystyle \lim_{x\rightarrow 0}\frac{\tan x-x}{x^3} = \frac{1}{3}$

But i did not understand how can i solve it without using $\displaystyle $ L hospital rule and series exp.

Help required, Thanks

Answer 1

$$=\dfrac{\sin\dfrac{\tan x+x}2}{\dfrac{\tan x+x}2}\cdot\dfrac{\dfrac{\tan x+x}2}{\dfrac x2} \cdot\dfrac{\sin\dfrac{\tan x-x}2}{\dfrac{\tan x-x}2}\cdot\dfrac{\tan x-x}{x^3}$$