I am computing a lot of cos(x)'s in my program. Is it more efficient in Fortran to stick with cos(x) or to calculate sqrt(1-sin(x)**2)? Will I be sacrificing any accuracy if the sqrt method is more efficient? And, I guess, the most important question if sqrt is more efficient. Can I expect that the compiler is recognizing this anyways and changing it for me?
The first thing you should note is that you'll have to be careful to pick the correct root for sqrt(1 - sin(x)**2), otherwise you run the risk of evaluating |cos(x)|, which is not the same. That itself adds complexity.
Use cos(x) as you should avoid such perceived micro-optimisations: you'll do well to beat any optimisation approach adopted by a modern FORTRAN compiler.
Even when I was using FORTRAN, the clever FORTRAN compilers would use the trig functions available on a chipset. sqrt is still on the whole implemented with a Newton-Raphson type algorithm and its evaluation will take a handful of clock cycles.
My hunch is that the latest compilers would reverse out your identity for you and substitute cos(x): check the output assembly.
But if you are in any doubt, profile it.