It is established etiquette by the Joint Chiefs of Staff, United States Department of Defense, the U.S. Army, the U.S. Navy, and the U.S. Air Force that the closing “Respectfully yours” be reserved for the President of the United States (according to U.S. Army standards, this also extends to the first lady and the President_elect).
In the end, this is all about respect for your recipient. If you’re offering a real solution to a real problem, people want to know about it. By following proper and well_understood formats and keeping your writing clear and engaging, you’re making it easier for them to solve their problem. Provided that you’re honest and have the skills and talents to back up your claims, writing a good, solid letter makes it more likely that both you and your reader will be satisfied. Try it the next time you have to write an important letter.