simplify the expression.
cos(2 tan^-1 x)
if tanθ = x, then cosθ = 1/√(1+x^2)
cos2θ = 2cos^2θ - 1 = 2/(1+x^2)-1 = (1-x^2)/(1+x^2)
To simplify the expression cos(2 tan^-1 x), we can use the double-angle identity for cosine.
The double-angle identity for cosine states that cos(2θ) = 1 - 2sin^2(θ).
In this case, let θ = tan^-1(x).
Thus, cos(2 tan^-1 x) = 1 - 2sin^2(tan^-1(x)).
Now, we need to find the value of sin(tan^-1(x)). To do that, we can use the identity for tangent inverse.
tan(tan^-1(x)) = x.
Since the tangent of the inverse tangent of x is equal to x, we can say x = sin(tan^-1(x))/cos(tan^-1(x)).
Since sin^2(θ) + cos^2(θ) = 1, we can find cos(tan^-1(x)) by using the Pythagorean identity.
cos(tan^-1(x)) = √(1 - sin^2(tan^-1(x))).
Now, substitute these values into the expression:
cos(2 tan^-1 x) = 1 - 2sin^2(tan^-1(x))
= 1 - 2[sin(tan^-1(x))/cos(tan^-1(x))]^2
= 1 - 2[x/√(1 - x^2)]^2
= 1 - 2x^2/(1 - x^2)
= (1 - x^2 - 2x^2)/(1 - x^2)
= (1 - 3x^2)/(1 - x^2).
Therefore, the expression cos(2 tan^-1 x) simplifies to (1 - 3x^2)/(1 - x^2).
To simplify the expression cos(2 tan^-1 x), we can use trigonometric identities.
Let's start by finding the expression for tan(2a) in terms of tan(a). The double-angle formula for tangent states:
tan(2a) = (2 tan(a))/(1 - tan^2(a))
Next, substitute a with arctan(x):
tan(2 tan^-1(x)) = (2 tan(arctan(x)))/(1 - tan^2(arctan(x)))
Since tan(arctan(x)) is simply x, we can simplify the expression further:
tan(2 tan^-1(x)) = (2x)/(1 - x^2)
Now, we need to find the expression for cos(2a) in terms of cos(a) and sin(a). The double-angle formula for cosine states:
cos(2a) = cos^2(a) - sin^2(a) = 1 - 2sin^2(a)
To simplify the expression cos(2 tan^-1(x)), we'll use the Pythagorean identity sin^2(a) + cos^2(a) = 1, re-arranging it to solve for sin^2(a):
sin^2(a) = 1 - cos^2(a)
Substituting this into the formula for cos(2a):
cos(2a) = 1 - 2(1 - cos^2(a))
Simplifying further:
cos(2a) = 2cos^2(a) - 1
Now, substitute a with tan^-1(x):
cos(2 tan^-1(x)) = 2cos^2(tan^-1(x)) - 1
Since cos^2(tan^-1(x)) is 1/(1 + x^2) (derived from the Pythagorean identity cos^2(a) + sin^2(a) = 1), we can simplify the expression:
cos(2 tan^-1(x)) = 2(1/(1 + x^2)) - 1
Final simplified expression: 2/(1 + x^2) - 1