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How would one prove that:


arctan x + arctan( (1)/(x) ) = (\pi )/(2)

what would happen if x < 0?

User Mig
by
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1 Answer

11 votes

Answer: If x<0, the identity does not always hold.

Explanation:

If
0 < \theta < (\pi)/(2), let
\arctan x=\theta, implying
x=\tan \theta, and vice versa.

We know that
\tan \left((\pi)/(2)-\theta \right)=\cot \theta=(1)/(\tan \theta), so this means that if we let
\tan \theta=x


\tan \left((\pi)/(2) -\theta \right)=(1)/(x)\\(\pi)/(2)-\theta=\arctan \left((1)/(x) \right)\\\theta+\arctan \left((1)/(x) \right)=(\pi)/(2)

Substituting back, we obtain that
\arctan (x)+\arctan \left((1)/(x) \right)=(\pi)/(2), as required.

User KubiRoazhon
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