arctan(x) + arctan(y) + arctan(z) = arctan\(\frac{x + y + z - xyz}{1 - xy - yz - zx}\)

We will learn how to prove the property of the inverse trigonometric function arctan(x) + arctan(y) + arctan(z) = arctan\(\frac{x + y + z - xyz}{1 - xy - yz - zx}\) (i.e., tan\(^{-1}\) x + tan\(^{-1}\) y + tan\(^{-1}\) z = tan\(^{-1}\) \(\frac{x + y + z - xyz}{1 - xy - yz - zx}\))

Prove that, tan\(^{-1}\) x + tan\(^{-1}\) y + tan\(^{-1}\) z = tan\(^{-1}\) \(\frac{x + y + z – xyz}{1 – xy – yz – zx}\)

 Proof : 

 Let, tan\(^{-1}\) x = α, tan\(^{-1}\) y = β and tan\(^{-1}\)γ

Therefore, tan α = x, tan β = y and tan γ = z

We know that, tan (α + β + γ) = \(\frac{tan α + tan β + tan γ - tan α tan β tan γ}{1 - tan α tan β - tan β tan γ - tan γ tan α}\)

tan (α + β + γ)   = \(\frac{x + y + z – xyz}{1 – xy – yz – zx}\)

α + β + γ = tan\(^{-1}\) \(\frac{x + y + z – xyz}{1 – xy – yz – zx}\)

or, tan\(^{-1}\) x + tan\(^{-1}\) y + tan\(^{-1}\) z = tan\(^{-1}\) \(\frac{x + y + z – xyz}{1 – xy – yz – zx}\).                Proved.

Second method:

We can prove tan\(^{-1}\) x + tan\(^{-1}\) y + tan\(^{-1}\) z = tan\(^{-1}\) \(\frac{x + y + z – xyz}{1 – xy – yz – zx}\) in other way.

We know that, tan\(^{-1}\) x + tan\(^{-1}\) y = tan\(^{-1}\) \(\frac{x + y}{1 – xy}\)

Therefore, tan\(^{-1}\) x + tan\(^{-1}\) y + tan\(^{-1}\) z = tan\(^{-1}\) \(\frac{x + y}{1 – xy}\) + tan\(^{-1}\) z

 tan\(^{-1}\) x + tan\(^{-1}\) y + tan\(^{-1}\) z = tan\(^{-1}\) \(\frac{\frac{x + y}{1 – xy} + z}{1 -  \frac{x + y}{1 - xy } ∙ z}\)

tan\(^{-1}\) x + tan\(^{-1}\) y + tan\(^{-1}\) z = tan\(^{-1}\) \(\frac{x + y + z – xyz}{1 – xy – yz – zx}\).          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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