Maximizing Your Math: Simplifying The Trigonometric Expression Tan(Tan^-1(X))

tan(tan−1(x)) =__________for x in__________ .

x; (−∞,∞)

The given expression is:
tan(tan^-1(x))

The value of tan^-1(x) is the angle whose tangent is x, so if we define the angle as θ, then we have:
tan(θ) = x

Now, let’s evaluate tan(tan^-1(x)) using the identity:
tan(arctan(x)) = x

We substitute θ for arctan(x), so we have:
tan(tan^-1(x)) = tan(θ)

Using the identity:
tan(θ) = sin(θ) / cos(θ)

We can rewrite the expression as:
tan(tan^-1(x)) = sin(θ) / cos(θ)

Using the Pythagorean identity:
sin^2(θ) + cos^2(θ) = 1

We have:
cos^2(θ) = 1 – sin^2(θ)

Replacing this in the previous expression, we get:
tan(tan^-1(x)) = sin(θ) / √(1 – sin^2(θ))

Now, we need to find the value of the sine of the angle θ, which is the same as finding the value of x, since we know that:
tan(θ) = x

Assuming that θ is in the range [-π/2 , π/2], we have:
sin(θ) = x / √(1 + x^2)

Therefore, the final expression for the given equation is:
tan(tan^-1(x)) = x / √(1 – x^2)

for x in [-1,1].

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