What's wrong with this idea to solve for $x$?

Question

I'm a tad confused about whether it makes sense to solve simultaneous equations, if they aren't linear. Does it make sense to solve a system of nonlinear equations simultaneously?

For instance: $$ \begin{cases} 5 + \cos x = 14.5 \\ 2.5 + \sin x = 1 \end{cases} $$

Can the above system be solved for $x$?

What I would do typically is to isolate $\cos$ and $\sin$, square both sides, and rewrite $\cos^2$ as $1-\sin^2$, and get something on the LHS such as $2\sin^2x = \dots$

Take positive square roots, then take $\arcsin$, to get a value for $x$.

There seems to be a flaw with this method, though.

Answer 1

The system has no solution:

$$5+\cos x=14.5\implies \cos x=9.5\;$$

and likewise the other one...but $\;-1\le\cos x,\,\sin x\le1\;$ ...

Answer 2

Note that

$$5+\cos x = 14.5 \implies \cos x=9.5$$

$$2.5+\sin x = 1 \implies \sin x=-1.5$$

which are both impossible since $$-1\le \cos x \le 1$$ $$-1\le \sin x \le 1$$

We could consider

$$5+\cosh x = 14.5 \implies \cosh x=9.5$$

$$2.5+\sinh x = 1 \implies \sinh x=-1.5$$

which neither has solution since

$$\cosh^2 x-\sinh^2 x=9.5^2-(-1.5)^2 \neq 1$$