Funny thing: by accident I typed in the following

Plot[Exp[-x] == Cos[x], {x, -1, 10}, PlotLabel -> "Plot[Exp[-x]==Cos[x]]"]

Which is different from

Plot[{Exp[-x], Cos[x]}, {x, -1, 10},  PlotLabel -> "Plot[{Exp[-x],Cos[x]}]"]

Normally I would expect a == b to be a logical expression which is either True or False.

Any explanation for the strange behaviour discovered by accident?

Regards, Wolfgang

Must be a peculiarity of

$Version

(* Out[40]= "8.0 for Microsoft Windows (64-bit) (October 7, 2011)" *)

In 5.2 for Microsoft Windows x86 (64 bit) (June 20, 2005), which I still keep on my PC (you will remember; it is much faster doing Integrate than Version 8) I get the error message "Plot::plnr : Exp[-x] == Cos[x] is not a machine size real number at x = -1 ..." and an empty Plot which is correct.

Are you looking for solutions to that equation?

Solve[Exp[x] == Cos[x] && -10 <= x <= 10, x, Reals]

(* Out[90]= {{x -> 0}, {x -> 
   Root[{-E^#1 + Cos[#1] &, -7.8535932799712482001}]}, {x -> 
   Root[{-E^#1 + Cos[#1] &, -4.7212927588476862166}]}, {x -> 
   Root[{-E^#1 + Cos[#1] &, -1.29269571937339838117}]}} *)

Or an iterative process that finds a solution from a starting point. For that one can use e.g. Newton iterations.

FixedPointList[{#[[1]] - f[#[[1]]]/f'[#[[1]]], 
   f[#[[1]]]} &, {-.7, f[-.2]}]

(* Out[112]= {{-0.7, -0.1613358247632598}, {-2.51705989723243, \
-0.268256883493079}, {-0.7474281702634069, 
  0.8919329283107287}, {-2.007816769456799, -0.2598565433504163}, \
{-1.285388510652165, 
  0.5575233413007432}, {-1.292717416706861, -0.005005670608255908}, \
{-1.2926957195615, 0.00001490725795189052}, {-1.292695719373398, 
  1.292351226034327*10^-10}, {-1.292695719373398, \
-5.551115123125783*10^-17}, {-1.292695719373398, \
-5.551115123125783*10^-17}} *)