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When integrate the indefinite integral Cos[nx]Cos[kx] about x, where both k and n are positive integer, the result is Pi when n equals to k and 0 when n is unequal to k. However, the code

sol = Integrate[Cos[n*x]*Cos[k*x], x, -Pi, Pi, 
 Assumptions -> n ∈ Integers && k ∈ Integers && n > 0 && k > 0]

gives the result (k Sin[π k + π n] - n Sin[π k + π n] +
k Sin[π k - π n] + n Sin[π k - π n])/(k^2 - n^2).

And then use the Simplify function,

Simplify[sol, Assumptions -> n ∈ Integers && k ∈ Integers && n > 0 && k > 0]

gives the result 0. Shouldn't that Integrate returns a Piecewise function like Piecewise[Pi, n == k, 0, n != k] instead?

This is well know issue. One way to handle it is

Simplify[ sol, 
 Assumptions -> Element[n, Integers] && Element[k, Integers] && n > 0 && k > 0 && k != n]

(* 0 *)

And

Simplify[ Limit[sol, k -> n], 
 Assumptions -> Element[n, Integers] && Element[k, Integers] && n > 0 && k > 0 && k == n ]

(* Pi *)

See

should-integrate-detect-orthogonality-of-functions-in-the-integrand

And

What assumptions to use to check for orthogonality

And

should-integrate-have-given-zero-for-this-integral

And

proper-way-to-simplify-integral-result-in-mathematica-given-integer-constraints

And

usage-of-assuming-for-integration