Consider the function
$$f(x)=\frac{\sin(x)+\sec(x)+\tan(x)}{\cos(x)\csc(x)\cot(x)}$$
in the interval $x \in[-\frac{\pi}{3},\frac{\pi}{4}]$.
Find a combination of algebraic (not transcendental) numbers $a,b,c,d$ such that $y=ax^3+bx^2+cx+d$ is closest to $f(x)$.
I am not asking for the solution, but asking for some hints/useful formulas that will help me to solve this problem.
Any help would be really appreciated. THANKS!
Note that the expression can be rewritten as
$$f(x)=\frac{\sin^2x(\sin x\cos x + \sin x +1)}{\cos^3x}$$
which has a double root at $x=0$. Thus, parametrize the cubic polynomial $y$ as
$$y = ax^2(x+\frac ba)$$
Moreover, note that $-\frac ba$ is the root of $\sin x\cos x + \sin x +1 = 0$, which is approximately $ -0.575$. What remains is to fit the parameter $a$.