How do you evaluate $cos(tan^-1(3/7))$ without a calculator?

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Answer 1 · 2016-10-07T13:09:06

$costheta=+-7/sqrt58$ depending on
whether $theta$ is in $I$ quadrant or $III$ quadrant.

Let $tan^(-1)(3/7)=theta$ , then $tantheta=3/7$

Note that we are seeking $cos(tan^(-1)(3/7))$ ie. $costheta$

As $tantheta=3/7$ ,

$sec^2theta=1+(3/7)^2=1+9/49=58/49$

Hence $sectheta=+-sqrt58/7$

and $costheta=+-7/sqrt58$ depending on whether $theta$ is in $I$ quadrant or $III$ quadrant.

How do you evaluate cos(tan^-1(3/7))cos(tan^-1(3/7)) without a calculator?