What is the exact value of $cos(tan^-1 (2/3))$ ?

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Answer 1 · 2015-04-15T14:14:15

$tan^-1(2/3)$ is a $t$ in $(- pi/2, pi/2)$ with $tan t = 2/3$

$t$ is in quadrant 1 or 2, so $cost$ is positive.

The problem has become:
$tan t = 2/3$ and $cost > 0$ .Find $cost$ .

Use your favorite method for solving such problems.

It's hard to draw and label a triangle or unit circle here, so I'll use the identities:

$cost = 1/sect$ and $tan^2t + 1= sec^2t$

$sec^2t = (2/3)^2 + 1 =4/9 + 1 = 13/9$

So, $sect = sqrt13/3$ (Remember $cost > 0$ , so $sect > 0$ as well)

$cost = 3/sqrt13 = (3sqrt13)/13$

What is the exact value of cos(tan^-1 (2/3))cos(tan^-1 (2/3))?