### Instruction: Find the value of \[\cot(\cot^{-1}3+\cot^{-1}7+\cot^{-1}13+\cot^{-1}21).\].### Response: I notice that the argument of the cotangent function is a sum of four inverse cotangent functions.
I recall that there is a formula for the inverse cotangent of a sum of two numbers: \[\cot^{-1}(x)+\cot^{-1}(y) = \cot^{-1}\left(\frac{xy-1}{x+y}\right).\]
I wonder if I can use this formula to simplify the argument of the cotangent function.
I start by grouping the first two terms and the last two terms: \[\cot(\cot^{-1}3+\cot^{-1}7+\cot^{-1}13+\cot^{-1}21) = \cot\left((\cot^{-1}3+\cot^{-1}7)+(\cot^{-1}13+\cot^{-1}21)\right).\]
I apply the formula to each pair of terms: \[\cot\left((\cot^{-1}3+\cot^{-1}7)+(\cot^{-1}13+\cot^{-1}21)\right) = \cot\left(\cot^{-1}\left(\frac{3\cdot 7-1}{3+7}\right)+\cot^{-1}\left(\frac{13\cdot 21-1}{13+21}\right)\right).\]
I simplify the fractions inside the inverse cotangent functions: \[\cot\left(\cot^{-1}\left(\frac{3\cdot 7-1}{3+7}\right)+\cot^{-1}\left(\frac{13\cdot 21-1}{13+21}\right)\right) = \cot\left(\cot^{-1}\left(\frac{20}{10}\right)+\cot^{-1}\left(\frac{272}{34}\right)\right).\]
I reduce the fractions to lowest terms: \[\cot\left(\cot^{-1}\left(\frac{20}{10}\right)+\cot^{-1}\left(\frac{272}{34}\right)\right) = \cot\left(\cot^{-1}(2)+\cot^{-1}(8)\right).\]
I apply the formula again to the remaining two terms: \[\cot\left(\cot^{-1}(2)+\cot^{-1}(8)\right) = \cot\left(\cot^{-1}\left(\frac{2\cdot 8-1}{2+8}\right)\right).\]
I simplify the fraction inside the inverse cotangent function: \[\cot\left(\cot^{-1}\left(\frac{2\cdot 8-1}{2+8}\right)\right) = \cot\left(\cot^{-1}\left(\frac{15}{10}\right)\right).\]
I reduce the fraction to lowest terms: \[\cot\left(\cot^{-1}\left(\frac{15}{10}\right)\right) = \cot\left(\cot^{-1}\left(\frac{3}{2}\right)\right).\]
I notice that the cotangent and inverse cotangent functions cancel out, leaving me with the argument: \[\cot\left(\cot^{-1}\left(\frac{3}{2}\right)\right) = \frac{3}{2}.\]