What is the probability density function and cumulative distribution function of $x$

Question

What is the probaiblity density function and cumulative distribution function of $x$ (where $x\in [\dfrac{-\pi}{2},\dfrac{\pi}{2}]$) such that both $y_1=\sin x$ and $y_2=\cos x$ are uniformly distributed in $[-1,1]$?

Answer 1

Consider the event $A$, defined as $|y_1| \le \frac12,$ or equivalently, $y_1 \in \left[-\frac12,\frac12\right]$. Since $y_1$ is uniformly distributed on $[-1,1]$, clearly $P(A) = \frac12.$

But since $y_1^2 + y_2^2 = 1,$ clearly $|y_1| \le \frac12$ precisely when $|y_2| \ge \sqrt{\frac34}.$ This means that the event $A$ occurs just when $$y_2 \in \left[-1,-\sqrt{\frac34}\right] \cup \left[\sqrt{\frac34},1\right]$$ which must occur with probability $1 - \sqrt{\frac34} \approx 0.134$ if $y_2$ is uniformly distributed on $[-1,1]$.

We have now shown that in order to construct the desired distribution, we must simultaneously satisfy $P(A) = \frac12$ and $P(A) \approx 0.134.$ We cannot do that, so we cannot construct the desired probability distribution.