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
- General and Principal Values of sin\(^{-1}\) x
- General and Principal Values of cos\(^{-1}\) x
- General and Principal Values of tan\(^{-1}\) x
- General and Principal Values of csc\(^{-1}\) x
- General and Principal Values of sec\(^{-1}\) x
- General and Principal Values of cot\(^{-1}\) x
- Principal Values of Inverse Trigonometric Functions
- General Values of Inverse Trigonometric Functions
- arcsin(x) + arccos(x) = \(\frac{π}{2}\)
- arctan(x) + arccot(x) = \(\frac{π}{2}\)
- arctan(x) + arctan(y) = arctan(\(\frac{x + y}{1 - xy}\))
- arctan(x) - arctan(y) = arctan(\(\frac{x - y}{1 + xy}\))
- arctan(x) + arctan(y) + arctan(z)= arctan\(\frac{x + y + z – xyz}{1 – xy – yz – zx}\)
- arccot(x) + arccot(y) = arccot(\(\frac{xy - 1}{y + x}\))
- arccot(x) - arccot(y) = arccot(\(\frac{xy + 1}{y - x}\))
- arcsin(x) + arcsin(y) = arcsin(x \(\sqrt{1 - y^{2}}\) + y\(\sqrt{1 - x^{2}}\))
- arcsin (x) - arcsin(y) = arcsin (x \(\sqrt{1 - y^{2}}\) - y\(\sqrt{1 - x^{2}}\))
- arccos (x) + arccos(y) = arccos(xy - \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\))
- arccos(x) - arccos(y) = arccos(xy + \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\))
- 2 arcsin(x) = arcsin(2x\(\sqrt{1 - x^{2}}\))
- 2 arccos(x) = arccos(2x\(^{2}\) - 1)
- 2 arctan(x) = arctan(\(\frac{2x}{1 - x^{2}}\)) = arcsin(\(\frac{2x}{1 + x^{2}}\)) = arccos(\(\frac{1 - x^{2}}{1 + x^{2}}\))
- 3 arcsin(x) = arcsin(3x - 4x\(^{3}\))
- 3 arccos(x) = arccos(4x\(^{3}\) - 3x)
- 3 arctan(x) = arctan(\(\frac{3x - x^{3}}{1 - 3 x^{2}}\))
- Inverse Trigonometric Function Formula
- Principal Values of Inverse Trigonometric Functions
- Problems on Inverse Trigonometric Function
11 and 12 Grade Math
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