SOLUTION: Express tan(2 arccos(x)) without trig functions

Question 905367: Express tan(2 arccos(x)) without trig functions
Answer by harpazo(655) (Show Source): You can put this solution on YOUR website!
Let us use a letter to represent 2 acrcos(x).
How about A.
We now have tan(A).
Using the Pythagorean Theorem, we get x^2 + y^2 = 1^1.
x^2 + y^2 = 1
Solve for y.
y^2 = 1 - x^2
Taking the square root of both sides we are left with
y = sqrt{1 - x^2}.
In the right triangle, we know that y = sqrt{1 - x^2}.
What is tangent?
Tangent is opposite over adjacent.
tan(2 arccos(x)) = 2x/(sqrt{1 - x^2})
We cannot leave a radical in the denominator of a fraction.
We rationalize the denominator and the final answer is

tan(2 arccos(x)) = 2x*sqrt{1 - x^2}/(1 - x^2)

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