How do you evaluate $cos(tan^-1(1/2))$ without a calculator?

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Answer 1 · 2016-09-23T04:04:50

$+-2/sqrt5$

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let $tan^-1(1/2) =a => tan a=1/2$

$H=sqrt(2^2+1^2)=sqrt5$

$cos(tan^-1(1/2))=cos a = 2/H=2/sqrt5$

As tangent is positive in both the first and the third quadrants,
$=> cosa=+-2/sqrt5$

positive sign for $a$ in the first quadrant and negative sign for $a$ in the third quadrant.

How do you evaluate cos(tan^-1(1/2))cos(tan^-1(1/2)) without a calculator?