One-to-one between $\cos(x)$ and $\cos^2(x)$ from $(0,\frac {\pi} 2)$?

Question

Is there an one-to-one relationship between values of $\cos(x)$ and $\cos^2(x)$ from $(0,\frac {\pi} 2)$? It seems like there is, but I'm not sure how to prove it.

Answer 1

I think you mean if $\cos x$ and $\cos^2 x$ are 1-to-1:

$$\forall x \in \left(0,\frac{\pi}{2}\right)\;\; \cos x \geq 0 \implies \cos^2x=c \iff \cos x = \sqrt{c}$$

Since $\cos$ is 1-to-1 over that range, there is a bijection between $x$ and $\cos$. Similarly, for $x^2$ over that range.

The composition of 1-to-1 functions is 1-to-1 so $\cos^2 x$ is 1-to-1.