PDF of negative $\cos(X)$

Question

Let $Y = - \cos(X)$, then what will be the pdf? Please share if you have any idea.

If $Y = \cos(X)$, where $X$ is uniformly distributed in the interval $(0, 2 \pi]$, then the pdf is given by

$$f_Y(y) = \begin{cases} \frac{1}{\pi\sqrt{1-y^2}}&\text{if }y\in[-1,1],\\ 0&\text{otherwise} \end{cases} $$

Source.

Let $Y = - \cos(X)$, then what will be the pdf and explanation?

My intention is to discuss the impact of the minus sign $(- \cos(X))$. What is the opinion of the learned members? Is it same like $Y = \cos(X)$? If it have identical answer, then what is the physical signification?

Answer 1

This might work (feedback is welcome). I will ignore the first like of your post.

You are given (you are telling me) that if $X\sim\text{unif}(0,2\pi)$, then $Y = \cos X$ will have the density given above.

Now you ask for $Z = -\cos X$. But this is just $Z = -Y$, and so we can apply a one-to-one transformation. $Y = -Z$, and $$f_Z(z) = \frac{f_Y(-z)}{\left|\frac{dz}{dy}\right|_{-z}} = \frac{\frac{1}{\pi\sqrt{1-(-z)^2}}}{|-1|} = \frac{1}{\pi\sqrt{1-z^2}}.$$

Alternatively, you can integrate $$P(Z\leq z) = P(-\cos X\leq z)$$ to find the cdf of $Z$. You can try that on your own.