Name: Multiple Angle Formula for Sine Inverse Function
Uploaded: 2019-07-19
Duration: 6 min 56 s
Description: What is twice of sine inverse function? Can we develop a formula for it? How to solve the problem using this function? Check out this video to know more.

Question

If

\tan^{- 1} x + \tan^{- 1} y + \tan^{- 1} z = \frac{π}{2}

and

x + y + z = \sqrt{3}

, then show that

x = y = z

.

Accepted Answer

Given that, $\tan^{- 1} x + \tan^{- 1} y + \tan^{- 1} z = \frac{π}{2}$ and $x + y + z = \sqrt{3}$ .

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

$\Rightarrow \tan^{- 1} (\frac{x + y}{1 - x y}) + \tan^{- 1} z = \frac{π}{2}$

$\Rightarrow \tan (\tan^{- 1} (\frac{x + y}{1 - x y})) = \tan (\frac{π}{2} - \tan^{- 1} z)$

$\Rightarrow \tan^{- 1} (\frac{x + y}{1 - x y}) = \cot^{- 1} z$

We know that, $\cot^{- 1} x = \tan^{- 1} (\frac{1}{x})$

$\Rightarrow \tan^{- 1} (\frac{x + y}{1 - x y}) = \tan^{- 1} (\frac{1}{z})$

Now, applying $\tan$ on both sides

$\Rightarrow \tan [\tan^{- 1} (\frac{x + y}{1 - x y})] = \tan [\tan^{- 1} (\frac{1}{z})]$

We know that, $\tan (\tan^{- 1} x) = x$

$\Rightarrow (\frac{x + y}{1 - x y}) = (\frac{1}{z})$

$\Rightarrow x z + y z = 1 - x y$

$\Rightarrow x y + x z + y z = 1$ .....(i)

Given equation is $x + y + 2 = \sqrt{3}$

Squaring on both sides,

$(x + y + z)^{2} = 3$

$\Rightarrow x^{2} + y^{2} + z^{2} + 2 (x y + y z + z x) = 3$

From equation(i)

$\Rightarrow x^{2} + y^{2} + z^{2} + 2 (1) = 3$

$\Rightarrow x^{2} + y^{2} + z^{2} = 1$

$\Rightarrow x^{2} + y^{2} + z^{2} - 1 = 0$ .....(ii)

Put equation (i) in (ii)

$x^{2} + y^{2} + z^{2} - (x y + y z + z x) = 0$

$\Rightarrow 2 (x^{2} + y^{2} + z^{2}) - 2 (x y + y z + z x) = 0$

$\Rightarrow (x^{2} - 2 x y + y^{2} + y^{2} - 2 y z + z^{2} + z^{2} - 2 z x + x^{2}) = 0$

$\Rightarrow (x - y)^{2} + (y - z)^{2} + (z - x)^{2} = 0$

Here, $x - y = 0$ , $y - z = 0$ and $z - x = 0$

$\Rightarrow x = y, y = z, z = x$

$\Rightarrow x = y = z$

Hence, showed.

If $\tan^{- 1} x + \tan^{- 1} y + \tan^{- 1} z = \frac{π}{2}$ and $x + y + z = \sqrt{3}$ , then show that $x = y = z$ .

Important Questions on Inverse Trigonometric Functions

Learn and Prepare for any exam you want !

If tan-1x+tan-1y+tan-1z=π2 and x+y+z=3, then show that x=y=z.

Important Questions on Inverse Trigonometric Functions

Learn and Prepare for any exam you want !

If $\tan^{- 1} x + \tan^{- 1} y + \tan^{- 1} z = \frac{π}{2}$ and $x + y + z = \sqrt{3}$ , then show that $x = y = z$ .