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Hi,

Should we write from and to address in IELTS general writing task 1 letter?

Thank you.
No. I believe instructions clearly states how to begin your letter. However, you should end your letter with proper closing e.g. Sincerely, Regards etc.
If you think about it, writing from and to addresses might increase word count but doesn't really demonstrate person's English writing skills.
 
No it is not necessary. Start with Opening Salutation like Dear Sir, Dear John, Write body paragraphs and end with appropriate closing like Yours faithfully or similar. Refer it IELTSLiz Writing task 1 for more details. Thats the source I am using to prepare. Best of luck !!