Below I have an equation that I need to solve and evaluate for $f(x)$, so that I can find the rule for $f^{-1}$$(x)$.
Solve $$ \sin^2(f(x))+2x \sin(f(x))+x^2=0 $$
Here is my progress:
$\frac{\sin^2(f(x))}{x}+2\sin(f(x))+x=0$
$\frac{\sin(f(x))}{x}$= $-2-\frac{x}{\sin(f(x))}$
And... I got stuck, so any help is appreciated! Thanks!
$$\sin^2(f(x))+2x\sin(f(x))+x^2=0$$
$$(\sin(f(x)) +x)^2=0$$ $$\sin(f(x))=-x$$ $$f(x)=-\sin^{-1} x$$ $$ f^{-1}x = -\sin x$$