50HZ Appliance & 60HZ Power Supply and vice versa

A 50 HZ power supply is common used in Europe while a 60 HZ power supply is common used in USA, Typically most of the electrical loads have a frequency range they are compatible with and it is common to see that the range span is from 50 HZ to 60 Hz

?What is the difference between 50 HZ and 60 HZ power transformer

Transformer which made for 60 HZ  is smaller than transformer made for 50 HZ , but in some cases they are in the same size and in this case the 60 HZ version will use lower grade steel laminations which mean it will  be cheaper than the other

?Can we connect 60 HZ power transformer to 50 HZ power supply

If you make this the transformer will get much hotter than normal and easily reach to a dangerous temperature that will cause transformer failure, this is because the transformer primary number of turns is designed according to input voltage to input frequency ratio (V/F) so if the frequency is reduced then (V/F) ratio will increase and there will not be sufficient number of turns to prevent transformer from reaching core saturation, So transformer will draw more much current from the main getting more hotter leading to failure

?Can we connect 50 HZ power transformer to 60 HZ power supply

Yes, you can and you will rarely have a problem as operating a 50 HZ transformer on 60 HZ power supply reduces the drawn current and hence the transformer will run a little cooler  

?Can we connect 50HZ induction motor to 60HZ Power Supply or vice versa

If I connect a 60 HZ induction motor to a 50 HZ power supply, the motor will draw more current and run at lower speed and if I connect a 50 HZ induction motor to a 60 HZ power supply the motor will draw lower current and run at high speed and in the two cases the result will be excessive heat buildup inside the motor that strongly shortened motor operating life

?How can you safely operate a 60 Hz appliance at a 50 HZ power supply or vice versa

:You should follow the following steps

First: contact the appliance manufacturer as it may be possible to safely 

.operate your appliance with the manufacturer approval

Second: some manufacturers sell the same appliance in 50 HZ and 60 HZ 

.countries, so try to connect manufacturer as he may have a solution
Third: you can purchase 60 HZ sine wave power supply but this solution is generally very expensive and it will be more cost effective to replace the appliance than to buy the 60 HZ power supply