# 3 arctan(x) = arctan($$\frac{3x - x^{3}}{1 - 3 x^{2}}$$)

We will learn how to prove the property of the inverse trigonometric function 3 arctan(x) = arctan($$\frac{3x - x^{3}}{1 - 3 x^{2}}$$) or, 3 tan$$^{-1}$$ x = tan$$^{-1}$$ ($$\frac{3x - x^{3}}{1 - 3x^{2}}$$).

Proof:

Let, tan$$^{-1}$$ x = θ

Therefore, tan θ = x

Now we know that, tan 3θ = $$\frac{3 tan θ - tan^{3}θ}{1 - 3 tan^{2}θ}$$

⇒ tan 3θ = $$\frac{3x - x^{3}}{1 - 3x^{2}}$$

Therefore, 3θ = tan$$^{-1}$$ ($$\frac{3x - x^{3}}{1 - 3x^{2}}$$)

⇒ 3 tan$$^{-1}$$ x = tan$$^{-1}$$ ($$\frac{3x - x^{3}}{1 - 3x^{2}}$$)

or, 3 arctan(x) = arctan($$\frac{3x - x^{3}}{1 - 3x^{2}}$$).           Proved.

Inverse Trigonometric Functions

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