Simplifying a trigonometric sum

Question

I want to caculate Cos[θ] + Cos[2 θ]+ ... + Cos[n θ]

Sum[Cos[k θ], {k, 1, n}]

Cos[1/2 (1 + n) θ] Csc[θ/2] Sin[(n θ)/2]

But when I use Euler's formula

Re[Expand[ExpToTrig[Sum[E^(I k θ), {k, 1, n}]]]]

I get

 Im[Sin[θ]/(-1 + Cos[θ] + I Sin[θ])] - 
 Im[(Cos[n θ] Sin[θ])/(-1 + Cos[θ] + I Sin[θ])] - 
 Im[(Cos[θ] Sin[n θ])/(-1 + Cos[θ] + I Sin[θ])] + 
 Re[-(Cos[θ]/(-1 + Cos[θ] + I Sin[θ])) + 
   (Cos[θ] Cos[n θ])/(-1 + Cos[θ] + I Sin[θ]) - 
   (Sin[θ] Sin[n θ])/(-1 + Cos[θ] + I Sin[θ])]

Why doesn't Mathmatica simplify it?

Answer 1

The rule of thumb that I use is this: Put Re as close as possible to the expression that I want to obtain its real (or imaginary) part and then put the result inside ComplexExpand and not Expand. (same applied to Im).

ComplexExpand assumes variables in expression are all real, and hence can do more simplification than Expand.

So the pattern is Simplify[ ComplexExpand [ Re [ .... ] ] ]

Compare the following 3 cases, all using the same expression to start with:

exp = ExpToTrig[Sum[E^(I k \[Theta]), {k, 1, n}]]

Case 1, Re outside, ComplexExpand inside

 Simplify@Re@ComplexExpand@exp

Case 2, Re inside, Expand outside

 Simplify@Expand@Re@exp

Case 3, Re inside, ComplexExpand outside

 Simplify@ComplexExpand@Re@exp