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  3. . for this problem, you may find proposition 3.1.5

. For this problem, you may find Proposition 3.1.5 useful which in turn implies that tan...

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Advance Math

. For this problem, you may find Proposition 3.1.5 useful whichin turn implies that tan x is continuous whenever x is not an oddmultiple of ? 2 . Moreover, you can assume that sin x and cos x arepositive and negative on appropriate intervals. Let I := (? ? 2 , ?2 ). (a) Show that tan x is strictly increasing and, hence,injective on I. (b) Prove that lim x?? ? 2 + tan x = ?? and lim x??2 ? tan x = ? Use this to conclude that f(I) = R. You may findExercise 6 in Section 3.7 useful here. (c) Show that arctan x =tan?1 x maps R to I and is differentiable everywhere with d dxtan?1 x = 1 1 + x 2 for all x ? R. (d) Prove that limx?? tan?1 x =? 2 and lim x??? tan?1 x = ? ? 2

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