Align equations which already have aligned answers (alignment-ception)

Question

I'm trying to align equations with already aligned case by case answers in them

What I am currently getting is:

and what I need is to align all the equals signs throughout the function

The required excerpt from my LaTex is

\documentclass[10pt,a4paper]{article}
\usepackage[latin1]{inputenc} 
\usepackage{amsmath}
\usepackage{mathtools}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{csquotes}

\begin{document}

\begin{multline}
\langle \cos(m \omega x), \cos(n \omega x) \rangle =\frac{1}{T}\int_{-T/2}^{T/2}\cos({m \omega x}) \cos({n \omega x}) dx = 
\begin{cases}
0 & \text{if } {m \neq n }  \\ 
\frac{1}{2} & \text{if } {m = n} > 0 \\
1 & \text{if } {m = n} = 0 \\ 
\end{cases} \\
\langle \sin(m \omega x), \sin(n \omega x) \rangle =\frac{1}{T}\int_{-T/2}^{T/2}\sin({m \omega x}) \sin({n \omega x}) dx = 
\begin{cases}
0 & \text{if } {m \neq n }  \\ 
\frac{1}{2} & \text{if } {m = n} > 0 \\
1 & \text{if } {m = n} = 0 \\
\end{cases}\\
\langle \sin(m \omega x), \cos(n \omega x) \rangle =\frac{1}{T}\int_{-T/2}^{T/2}\sin({m \omega x}) \cos({n \omega x}) dx = 0
\end{multline}

\end{document}

I've tried using

\begin{aligned}[t]
one function with cases
\end{aligned}\\
\begin{algined}[t]
next function with cases
\end{aligned}\\
\begin{algined}[t]
next function without cases
\end{aligned}\\

inside the \begin{multline}...\end{multiline}, but I've had no success.

I assume it can be done given that I can see it in this paper, but I am very new to LaTex. It would be interesting to know how it is able to deal with this sort of alignment issue, given that for each equation, there are different amounts of equals signs, and therefore different amounts of alignments required.

Is there are way of replacing the active symbol & for aligned and not case?

Answer 1

There is a simple solution, instead of multiline directly use the align-environment, which is automatically in math-mode and allows various alignment operations. To simply align at the = sign, use the &-sign. You can also have additional &-signs for formatting.

 \documentclass[11pt]{article} 
\usepackage{amsmath}
\begin{document}
\begin{align}
\langle \cos(m \omega x), \cos(n \omega x) \rangle &=\frac{1}{T}\int_{-T/2}^{T/2}\cos({m \omega x}) \cos({n \omega x}) dx = 
\begin{cases}
0 & \text{if } {m \neq n }  \\ 
\frac{1}{2} & \text{if } {m = n} > 0 \\
1 & \text{if } {m = n} = 0 \\ 
\end{cases} \\
\langle \sin(m \omega x), \sin(n \omega x) \rangle &=\frac{1}{T}\int_{-T/2}^{T/2}\sin({m \omega x}) \sin({n \omega x}) dx = 
\begin{cases}
0 & \text{if } {m \neq n }  \\ 
\frac{1}{2} & \text{if } {m = n} > 0 \\
1 & \text{if } {m = n} = 0 \\
\end{cases}\\
\langle \sin(m \omega x), \cos(n \omega x) \rangle &=\frac{1}{T}\int_{-T/2}^{T/2}\sin({m \omega x}) \cos({n \omega x}) dx = 0
\end{align}
\end{document}