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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