Simplify the expression:
cos(2tan^(-1)x)
any help is appreciated =D
tan^-1 x = arctan x are the same thing
The angle "A" with a tangent of x has sin A = x/sqrt(x^2+1) and
cos A = 1/sqrt(x^2+1)
The cosine of twice that angle is
cos^2 A - sin^2 A = (1 - x^2)/(1 + x^2)
(1-x^2)/(1+x^2) is the answewr
Check: If x = 1,
A = tan^-1x = 45 degrees
2A = 90 degrees
cos 90 = 0, which agrees with (1-x^2)/(1+x^2)
To simplify the expression cos(2tan^(-1)x), we can use a trigonometric identity which states that cos(2θ) can be expressed in terms of tan(θ).
Let's break down the steps to simplify the expression:
Step 1: Identify the trigonometric identity
The identity we will use is cos(2θ) = 1 − 2sin²θ.
Step 2: Determine θ
In our expression, θ is tan^(-1)x.
Step 3: Substitute θ into the identity
cos(2tan^(-1)x) = 1 − 2sin²(tan^(-1)x).
Step 4: Use an additional identity
Since sin²(θ) + cos²(θ) = 1, we can rewrite sin²(θ) as 1 − cos²(θ).
Step 5: Substitute the new identity into the expression
cos(2tan^(-1)x) = 1 − 2(1 − cos²(tan^(-1)x)).
Step 6: Simplify further
cos(2tan^(-1)x) = 1 − 2 + 2cos²(tan^(-1)x).
Step 7: Combine like terms
cos(2tan^(-1)x) = -1 + 2cos²(tan^(-1)x).
Thus, after simplification, the expression cos(2tan^(-1)x) can be written as -1 + 2cos²(tan^(-1)x).