Alignment in aligned and aligned[t] environments

Question

The following code instructs LaTeX to typeset two different expressions for the square of the absolute value of a certain sum. I have three alignments - one of them is "almost" correct and two of them are correct.

The "=" are aligned properly. Though, the spaces to the right of the two equal signs are different. (This should shift the other alignments, but it doesn't.) I think the space to the right of the second equal sign is too little. How do I correct that?

I would like the first "+" in lines 2, 4, and 6 to be aligned. It seems that if LaTeX did not put an unreasonable amount of space after the "3" in line 2, these plus signs would have been aligned. In the aligned[t] environment, I have two alignments. It seems that since I have "3 &&+" to begin coding for line 2, the second "&" artificially put in the extra space.

The third alignment is to align cos(2a + b + c) in line 2, the first "+" in line 3, cos(a - b) in line 4, and the first "+" in line 5. This is correct. (I wanted to say that to help those looking at the code.)

\begin{align*}
\MoveEqLeft[3] \vert xy + xz + yz \vert^{2} \\
&= \begin{aligned}[t] 3 &&+ \bigl[&\cos(2a + b + c) + \cos(a + 2b + c) + \cos(a + b + 2c) \\
&&& + \cos(a - b) + \cos(a - c) + \cos(b - c)\bigr] \\
&&+ \bigl[&\cos(a - b) + \cos(a - c) + \cos(b - c) \\
&&&- \bigl(\cos(2a + b + c) + \cos(a + 2b + c) + \cos(a + b + 2c)\bigr) \bigr] \\
\end{aligned} \\
&= 3 + 2\bigl(\cos(a - b) + \cos(a - c) + \cos(b - c)\bigr) .
\end{align*}

Answer 1

Here is a suggestion using a combination of alignat* and \mathrlap

\documentclass{scrartcl}
\usepackage{mathtools}% loads also amsmath
\begin{document}
\begin{alignat*}{3}
  \MoveEqLeft[3] \mathrlap{\vert xy + xz + yz \vert^{2}} \\
  &= 3&& + \bigl[&&\cos(2a + b + c) + \cos(a + 2b + c) + \cos(a + b + 2c) \\
  &&&&&+ \cos(a - b) + \cos(a - c) + \cos(b - c)\bigr] \\
  &&&+ \bigl[&&\cos(a - b) + \cos(a - c) + \cos(b - c) \\
  &&&&&- \bigl(\cos(2a + b + c) + \cos(a + 2b + c) + \cos(a + b + 2c)\bigr) \bigr] \\
  &= 3&&\mathrlap{{}+ 2\bigl(\cos(a - b) + \cos(a - c) + \cos(b - c)\bigr).} 
\end{alignat*}
\end{document}

Or maybe better

\documentclass{scrartcl}
\usepackage{mathtools}% loads also amsmath
\begin{document}
\begin{alignat*}{3}
  \vert xy + xz + yz \vert^{2}
      &= 3&& + \bigl[&&\cos(2a + b + c) + \cos(a + 2b + c) + \cos(a + b + 2c) \\
  &&&&&+ \cos(a - b) + \cos(a - c) + \cos(b - c)\bigr] \\
  &&&+ \bigl[&&\cos(a - b) + \cos(a - c) + \cos(b - c) \\
  &&&&&- \bigl(\cos(2a + b + c) + \cos(a + 2b + c) + \cos(a + b + 2c)\bigr) \bigr] \\
  &= 3&&\mathrlap{{}+ 2\bigl(\cos(a - b) + \cos(a - c) + \cos(b - c)\bigr).} 
\end{alignat*}
\end{document}

\mathrlap is provided by the mathtools package. It 'hides' the width of its argument and so the last equation line does not influence the 4th and 5th &.

The inserted space around a + depends on its current meaning. The {}+ in the last line shows latex that the + represents an addition and not only a positive number.

Answer 2

I think it would look better with multlined: the formula is easier to read, has a simpler syntax, and the 3+ signs are aligned!:

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[showframe, nomarginpar]{geometry}
\usepackage{amssymb,mathtools}
\usepackage{amsfonts}

\begin{document}

    \begin{align*}
     \vert xy + xz + yz \vert^{2} = 3 & +\!\! \begin{multlined}[t]
                     \bigl[\cos(2a + b + c) + \cos(a + 2b + c) + \cos(a + b + 2c) \\
                    + \cos(a - b) + \cos(a - c) + \cos(b - c)\bigr]
                \end{multlined} \\
    & + \!\!\begin{multlined}[t]
     \bigl[\cos(a - b) + \cos(a - c) + \cos(b - c) \\
    - \bigl(\cos(2a + b + c) + \cos(a + 2b + c) + \cos(a + b + 2c)\bigr) \bigr]
                \end{multlined} \\
    = 3 &+ 2\bigl(\cos(a - b) + \cos(a - c) + \cos(b - c)\bigr) .
    \end{align*}

\end{document} 

Concerning the different spacing on the right of the = signs, it's the first one that is a little too large. If you a negative thin space (\!), it's OK.