If you hold a constant CAS and the air temperature changes (let's say it gets colder) then the air density changes. The air will get thicker. So for the same CAS, your TAS reduces.
However, Mach No is TAS/local speed of sound, and the speed of sound is dependent also on the temperature. As the temperature decreases, the LSS also decreases and the ratio TAS/LSS (ie, your Mach No) remains the same.
At a given pressure level, the ratio of CAS to Mach No is not affected by temperature.
As for formulae, there are many and they vary between very precise, but not easy to use in the air, to being quick to calculate, but less accurate.
Quite a useful one for TAS (up to about 25000 feet) is TAS = CAS + 1.75% per thousand feet. For instance, if you are at 10000 feet with a CAS of 100 knots, add 10 lots of 1.75%, which gives 17.5%. The TAS is 117.5 knots. At ISA temp, it gives an answer within 1 knot.
For a detailed list of formulae, try
this link