My Sony 42" LED TV screen is not working only black video but it works a audio, what is the problem? And How can I fix?

Sorry to tell you but unless you are an electronics tech and have the equipment and schooling... you aren't going to be doing this yourself.

This comes from someone that DOES have an Engineering degree in Electronics... even if it was mine... by the time I bought schematics and took the time to locate the fault (even without factoring the value of my time) its possible the failed module may not be cost effective to repare the unit.

Your options are take it to a repair shop or replace it. If its more than a couple years old.. odds are replacing it will be cheaper.

I would also try different sources and inputs.

And double check your cables too.

Especially the Video cable... I've had even high quality cables fail before (Monster Cable). Don't assume they are good.