Use a right triangle to rewrite as an algebraic expression c

Use a right triangle to rewrite as an algebraic expression: cos(arctan(x/2))

Solution

Let y=arctan(x)
Then:
tan y = x
so sec^2 y = x^2 + 1( as sec^2 y = tan ^2y + 1)

so cos^2 y = 1/(x^2+1) or cos y = 1/sqrt(x^2+1)
so sin y = tan y cos y = x/sqrt(x^2+1)



cos(2arctan(x)) =
cos(2y) =
cos^2(y) - sin^2(y) =
1/(x^2+1) - x^2/(x^2+1) =
(1-x^2)/(1+x^2)