Question

find the value of cos (tan inverse 2)

Answer 1

Easy way to do is to draw a triangle with tan theta = 2.

See the attachment.

If tan theta = 2

altitude = 2
base = 1
hypotenuse = √5

Cos (tan inverse 2) = 1/√5

Answer 2

$\cos( { \tan }^{ - 1} (2)) \\ here \: let \: tan \alpha = 2 \\ \cos \alpha = \frac{1}{ \sqrt{5} } \\ \alpha = {tan}^{ - 1} (2) \\ \alpha = { \cos }^{ - 1} ( \frac{1}{ \sqrt{5} } ) \\ { \cos }^{ - 1} ( \frac{1}{ \sqrt{5} } ) = {tan}^{ - 1} (2) \\ now \\ \cos( { \tan }^{ - 1} (2)) \\ = \cos( { \cos}^{ - 1}( \frac{1}{ \sqrt{5} } )) \\ = \frac{1}{ \sqrt{5} }$
hope it helps