Is there an $x$ such that $\cos(x)=\sin(x)=0$?

Question

How do I show this in proofs? All I know is that there is no one value of $x$ that will make both $\cos x$ and $\sin x$ equal to $0$, but I'm unsure how to prove it other than mentioning what I just said.

Let $f(x)=\sin x$ and $g(x)=\cos x$. Let $D$ be the collection of real numbers and consider the propositional functions with the domain of definition $D$ defined by $P(x):f(x)=0$ and $Q(x):g(x) = 0$ when $x$ is a real number. In other words for a real number $x$, $P(x)$ is the proposition $f(x) = 0$ and $Q(x)$ is defined similarly.

Let $R$ be the statement $∃x, P(x)∧Q(x)$. Is $R$ true or false?

Answer 1

Suppose there was a point $x\in\mathbb{R}$ such that

$$\cos x=\sin x=0.$$

Then we would have

$$0=\cos^2x+\sin^2x=1,$$

which is a contradiction. Thus such a point cannot exist.