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Verify the following Machin-like formulas are correct by calculating the value of '''tan'''(''right hand side)'' for each equation using exact arithmetic and showing they equal 1:
: <math>{\pi/4\over4} = \arctan({1/2)\over2} + \arctan({1/3)\over3}</math>
: <math>{\pi/4\over4} = 2\times \arctan({1/3)\over3} + \arctan({1/7)\over7}</math>
: <math>{\pi/4\over4} = 4\times \arctan({1/5)\over5} - \arctan({1/239)\over239}</math>
: <math>{\pi/4\over4} = 5\times \arctan({1/7)\over7} + 2\times \arctan({3/79)\over79}</math>
: <math>{\pi/4\over4} = 5\times \arctan({29/278)\over278} + 7\times \arctan({3/79)\over79}</math>
: <math>{\pi/4\over4} = \arctan({1/2)\over2} + \arctan({1/5)\over5} + \arctan({1/8)\over8}</math>
: <math>{\pi/4\over4} = 4\times \arctan({1/5)\over5} - \arctan({1/70)\over70} + \arctan({1/99)\over99}</math>
: <math>{\pi/4\over4} = 5\times \arctan({1/7)\over7} + 4\times \arctan({1/53)\over53} + 2\times \arctan({1/4443)\over4443}</math>
: <math>{\pi/4\over4} = 6\times \arctan({1/8)\over8} + 2\times \arctan({1/57)\over57} + \arctan({1/239)\over239}</math>
: <math>{\pi/4\over4} = 8\times \arctan({1/10)\over10} - \arctan({1/239)\over239} - 4\times \arctan({1/515)\over515}</math>
: <math>{\pi/4\over4} = 12\times \arctan({1/18)\over18} + 8\times \arctan({1/57)\over57} - 5\times \arctan({1/239)\over239}</math>
: <math>{\pi/4\over4} = 16\times \arctan({1/21)\over21} + 3\times \arctan({1/239)\over239} + 4\times \arctan({3/1042)\over1042}</math>
: <math>{\pi/4\over4} = 22\times \arctan({1/28)\over28} + 2\times \arctan({1/443)\over443} - 5\times \arctan({1/1393)\over1393} - 10\times \arctan({1/11018)\over11018}</math>
: <math>{\pi/4\over4} = 22\times \arctan({1/38)\over38} + 17\times \arctan({7/601)\over601} + 10\times \arctan({7/8149)\over8149}</math>
: <math>{\pi/4\over4} = 44\times \arctan({1/57)\over57} + 7\times \arctan({1/239)\over239} - 12\times \arctan({1/682)\over682} + 24\times \arctan({1/12943)\over12943}</math>
: <math>{\pi/4\over4} = 88\times \arctan({1/172)\over172} + 51\times \arctan({1/239)\over239} + 32\times \arctan({1/682)\over682} + 44\times \arctan({1/5357)\over5357} + 68\times \arctan({1/12943)\over12943}</math>
and confirm that the following formula is incorrect by showing '''tan'''(''right hand side)'' is ''not'' 1:
: <math>{\pi/4\over4} = 88\times \arctan({1/172)\over172} + 51\times \arctan({1/239)\over239} + 32\times \arctan({1/682)\over682} + 44\times \arctan({1/5357)\over5357} + 68\times \arctan({1/12944)\over12944}</math>
These identities are useful in calculating the values:
: <math>\tan(a + b) = ({\tan(a) + \tan(b))/( \over 1 - \tan(a)* \tan(b))}</math>
: <math>\tan\left(\arctan({a/\over b)}\right) = {a/\over b}</math>
: <math>\tan(-a) = -\tan(a)</math>
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