Inconsistencies with multiple differentiation methods

Question

$$w=\sin x$$

$$\frac{dw}{dx} = \cos x$$

$$\therefore\frac{dx}{dw} = \frac{1}{\cos x}$$

Rearranging the initial relationship;

$$x = \arcsin(w)$$

$$\therefore\frac{dx}{dw} = \frac{1}{(1-w^2)^{0.5}}$$

But,

$$\frac{1}{\cos x} \neq \frac{1}{(1-w^2)^{0.5}}$$

What's wrong with one of the methods?

Answer 1

$\dfrac{1}{\cos x}$ is indeed equal to $\dfrac{1}{(1-w^2)^{0.5}}$ when $\cos x$ is positive.