Question

How do you solve and find the value of `cos(tan^-1sqrt3)`?

Answer 1

`cos(tan^(-1)(sqrt(3)))=color(green)(1/2)`

Explanation:

The standard definition of the inverse tan function implies an angle either Quadrant I or II.

The angle `theta=tan^(-1)(sqrt(3))` can be represented by means of a right triangle in standard position with opposite side of length `sqrt(3)` and adjacent side of length `1`.
By the Pythagorean Theorem this implies a hypotenuse of `sqrt((sqrt(3)^2+1^2))=sqrt(4)=2`

Using this triangle and the definition of `cos` as `"adjacent"/"hypotenuse"`
we have
`color(white)("XXX")cos(tan^(-1)(sqrt(3)))=1/2`