This is just a variation of a problem I've seen given many times. It is one of the simplest equations I can think of that has no analytical solution. Think of it as due to the presence of x inside and out of the trig functions.
First, a simple re-write of the problem. We can divide by cos(x), because (n+1/2)*pi is never a solution, and that is where cos(x) is equal to zero. But now we have the problem
-x = tan(x)
First, wheredo the solutions lie? We can see that by plotting the two sides of that equation.
fplot(@(x) -x,[-20,20],'r');
legend('LHS: y = -x','RHS: y = tan(x)')
Every place where the red and blue lines cross is a solution. Do you see there will be infinitely many solutions? In fact, the solutions will be roughly separated by increments of pi. However, there is no analytical solution to be found, even in this form.
solve(-X == tan(X),'returnconditions',true)
Warning: Unable to find explicit solution. For options, see
help.
ans =
X: [0×1 sym]
parameters: [1×0 sym]
conditions: [0×1 sym]
Tools like fzero, or vpasolve can solve it numerically of course. (fzero is a better choice always than fsolve, because fzero is designed to handle single variable problems very well, whereas fsolve is targeted at multidimensional problems.)
So how can we find the n'th positive solution numerically? This is where fzero is perfect, because we can provide an interval for the solution. And we know an interval where the nth root will lie. That is, if we call the zeroth root, the simple solution at x==0, then the nth root must lie in the interval pi*[n-1/2,n+1/2].
tanfunroot = @(n) fzero(@(x) tan(x) + x,pi*[n - 1/2 + eps(n),n + 1/2 - eps(n)]);
Note the way I have carefully tweaked the ends of the interval, to insure that fzero will have no problems.
It will even generate the negative roots, though you might have seen that if X is a root of the problem, then -X is also a root.
Are these solutions to the original problem? Of course they are.
Finally, to aid your intuition that successive roots will be separated by approximately pi, try this:
tanfunroot(100) - tanfunroot(99)
Pretty close to pi. That is generally true, except for the roots nearest to zero. And the farther away from zero you go, the increment will becomes closer and closer to pi.
