Does a solution exist for EVERY possible function you can write down? (NO.) In fact, it is trivially easy to write down infinitely many sets of equations that have no solution.
syms x y
A=(1-sqrt(1-x))^.5*tan((1-sqrt(1-x))^.5*y);
B=(1+sqrt(1-x))^.5*tan((1+sqrt(1-x))^.5*y);
A - B
I'm not sure why you would expect to see an algebraic solution to that problem.
fimplicit(A-B)
Each blue line in that solution is an entire family of solutions to the problem. So, given any value for y, there are infinitely many possible solutions for x, given y. But there will be no simple algebraic solution, nothing nice and simple you can write down.
In general (I said this just the other day too) when you have a variable both inside and outside a trig function, you will never find an analytical solution. (There will surely be some rare counter-examples to that statement, but it is true in general.)
