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Jul 31, 2012
Problem. Show that the following are all equal to each other:
tan(99 degrees), tan(999 degrees), tan(9999 degrees), tan(99999 degrees), etc.
Hint/Proof-Outline. It suffices to show that each of these is equal to tan(99 degrees). Now observe that the equation "tan(999...9 degrees) = tan(99 degrees)" is equivalent to an equation of the form AB - CD = 0, where A,B,C,D are the sine or cosine of 99 degrees or 999...9 degrees. Use the sine angle subtraction formula to compress this and then note that the angle you're taking the sine of is an integer multiple of 180 degrees. Finally, work backwards from this to get a proof of what we wanted to prove.
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There is an easier way to show this than the way I suggested earlier.
999 = 900 + 99 = 5(180) + 99,
9999 = 9000 + ...See More