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Question

Prove that tan1x<x, for all x>0

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Solution

Let f(x)=tan1(x)x
f(x)=11+x21=11x21+x2
=x21+x2=<0 for all x>0.
f(x) is strictly decreasing for all x>0.
f(x)<f(0) for all x>0.
tan1xx<tan1(0)0 for all x>0
tan1(x)x<00 for all x>0
tan1(x)x<0 for all x>0
tan1(x)<x for all x>0
Hence, proved.

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