Mathematica – Useful Functions

Extract coefficients from symbols Sin[x 2] and Cos[x 2]

 Coefficient[2 Sin[x 2] + Cos[x 2], #] & /@ {Sin[x 2], Cos[x 2]}

Find derivative of some exp

part1D0 = d1 Sin[\[Beta] x] + d2 Cos[\[Beta] x] /. {\[Beta] -> 2};
part1D1 = D[part1D0, x]; (* Derivative of part1D0 *)

Combine two expressions into system of linear equations

lhs = {2 a1 , 1 a2 , 3 a3};
rhs = {1,5,7};
system = Thread[ lhs == rhs ];
solutions = Solve[system, {a1, a2, a3} ]

Clear all global variables

ClearAll["Global`*"];

Collecting terms

Collect[a1  Sin[\[Beta] x] + a2  Sin[\[Beta] x] + a1  Cos[\[Beta] x] - 
  a2  Cos[\[Beta] x], {Sin[\[Beta] x], Cos[\[Beta] x]}]
(* Will give a1 - a2) Cos[x \[Beta]] + (a1 + a2) Sin[x \[Beta]] *)