By means of sketches of the graphs y = 1/x and y = tan x, show that

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By means of sketches of the graphs y = 1/x and y = tan x, show that the equation x tan x = 1 has a root between x = 0 and x = 1/2π and an infinity of roots near x = kπ, where k = 1, 2, 3, . . . . . Deduce which of the two iterations

(a) xn + 1 = cot xn 

(b) xn + 1 = tan−1(1/xn) + kπ

is convergent to the roots, and use it to locate the smallest positive root to 6dp.

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