`arctanx + arctan(1/x) = pi/2 , x>0` Verify each identity

Wednesday, August 24, 2016

$\displaystyle{\arctan{{x}}}+{\arctan{{\left(\frac{{1}}{{x}}\right)}}}=\frac{\pi}{{2}},{x}>{0}$ Verify each identity

We will use the following formula:

$\displaystyle{\arctan{{\left(\frac{{1}}{{x}}\right)}}}=\frac{\pi}{{2}}-{\arctan{{\left({x}\right)}}},$ if $\displaystyle{x}>{0}$

If we apply the above formula to the left side of our expression we get

$\displaystyle{\arctan{{\left({x}\right)}}}+{\arctan{{\left(\frac{{1}}{{x}}\right)}}}={\arctan{{\left({x}\right)}}}+\frac{\pi}{{2}}-{\arctan{{\left({x}\right)}}}=\frac{\pi}{{2}}$

Q.E.D.

One should note that the formula from the beginning holds only for positive numbers. For negative numbers we have a slightly different formula holds.

$\displaystyle{\arctan{{\left(\frac{{1}}{{x}}\right)}}}=-\frac{\pi}{{2}}-{\arctan{{\left({x}\right)}}},$ if $\displaystyle{x}<{0}$

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Wednesday, August 24, 2016

$\displaystyle{\arctan{{x}}}+{\arctan{{\left(\frac{{1}}{{x}}\right)}}}=\frac{\pi}{{2}},{x}>{0}$ Verify each identity

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Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

Wednesday, August 24, 2016

Verify each identity

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Post a Comment

Thomas Jefferson&#39;s election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

$\displaystyle{\arctan{{x}}}+{\arctan{{\left(\frac{{1}}{{x}}\right)}}}=\frac{\pi}{{2}},{x}>{0}$ Verify each identity

Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?