NEED ASAP! use a right triangle to write cos(tan^-1 x/2). as an algebraic expression

Question

cos ( tan ^{-1 x}/₂). (x over 2)

Answer 1

The right triangle has opposite side of length x, adjacent side of length 2.

From the Pythagorean theorem, the hyp has length sqrt(4 + x^2 )

From there, the cosine is 2 / sqrt( 4 + x^2 )

Answer 2

You take tan^-1 of two sides of a triangle, the opposite side divided by the adjacent side (so "x" divided by "2" means "x" is the opposite side and "2" is the adjacent side), and the result is an angle. So that would look like this (where θ is the angle):

Then, you can use the Pythagorean Theorem to solve for the hypotenuse:

2² + x² = (hypotenuse)²

hypotenuse = √(x² + 4)

Now, since cosine is the adjacent side divided by the hypotenuse, that would be 2/√(x² + 4)

So cos(tan^-1(x/2) = 2/√(x² + 4)

If you are required to, you can rationalize the denominator by multiplying by √(x² + 4)/√(x² + 4) to get: