Find lim x-->0 sinx-tanx / x^3 PLEASE HELP

final answer is -1/2 please show me the full steps, I am so lost.

Without using L'Hopital's Rule:

lim(x→0) (sin x - tan x)/x^3
= lim(x→0) (sin x - sin x/cos x)/x^3
= lim(x→0) (sin(x)/x) * ((1 - 1/cos x)/x^2)
= lim(x→0) (sin(x)/x) * (1/cos x) * ((cos x - 1)/x^2)
= lim(x→0) (sin(x)/x) * (1/cos x) * ((cos x - 1)*(cos x + 1)/(x^2 * (cos x + 1)))
= lim(x→0) (sin(x)/x) * (1/cos x) * ((cos^2(x) - 1)/(x^2 * (cos x + 1)))
= lim(x→0) (sin(x)/x) * (1/cos x) * -sin^2(x)/(x^2 * (cos x + 1))), via trig. identity
= lim(x→0) (sin(x)/x)^3 * (1/cos x) * (-1/(cos x + 1))
= 1^3 * (1/1) * (-1/2)
= -1/2.

I hope this helps!