A little more than the problem you are trying to solve:

• http://mathworld.wolfram.com/TancFunction.html

If you follow the SE article carefully, you will have a very nice conclusion (though you have Cotc, the argument is the same)

• https://math.stackexchange.com/a/451723/111285

Therefore, you have a numerically robust way to find nth solution when n is very large:

In[1]:= n=100;
In[2]:= FindRoot[Cot[x]==x,{x,n*\[Pi]+1/(n*\[Pi])}]
Out[2]= {x->314.162}

A notebooks is attached for you to play around.

Attachments:

solution.nb

Try restricting the range of the variable:

Solve[Cos[x] - x*Sin[x] == 0 && Abs[x] < 10, x]