3 arctan(x) | 3 tan(^{-1}) x |3 tan inverse x

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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

11 and 12 Grade Math

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