Install packages on cygwin

If you want LATEX in Cygwin so you can use Bash Script and commands line you may choose to install a complete version of for example Miktex and Texniccenter in Windows and then create a symbolic link to the .exe files you need to compile (e.g. latex.exe, pdflatex.exe, ...) your .tex file from Cygwin Bash shell:

ln -s 'C:\Program Files (x86)\MiKTeX 2.9\miktex\bin\x64\latex.exe'  /usr/local/bin/latex

I have not used Cygwin for a long time but the last time I checked the default TeX installation was still teTeX. I do not use tikz so I do not know if it was part of teTeX. In any case if you really want you could try to install TeXLive. In theory that should not be a problem but in practice that is another story. You can definitely use TeXLive in Cygwin from live USB or CD. The real question for me is why do you need Cygwin to use tikz on your Windows machine when you have native MiKTeX package of TeX and friends for Windows as well as TeXLive which in spite of being more geared towards Unix and Unix like system can be easily installed natively? I am guessing that teTeX is needed just as a dependency for some other Cygwin packages which are not natively available for Windows. Now historically speaking installing TeX on Windows for example 10 years ago (before ProTeXt MiKTeX installer) was not actually trivial and Cygwin route was the simplest one.