If tan-1x-tan-1y-tan-1z- - then x-y-z-

Explanation for the correct option:Use formula tan^ 1A tan^ 1B=tan^ 1(A B/1+AB)Given equation tan^ 1x+tan^ 1y+tan^ 1z=π , then [ tan^ 1x+tan^ 1y=π tan^ 1z; ta ...

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Standard XII

Mathematics

Properties Derived from Trigonometric Identities

If tan-1x+tan...

Question

If $\tan^{- 1} x + \tan^{- 1} y + \tan^{- 1} z = π,$ then $x + y + z =$

$x y z$

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$0$

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$1$

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$2 x y z$

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Solution

The correct option is A
$x y z$

Explanation for the correct option:
Use formula $\tan^{- 1} A - \tan^{- 1} B = \tan^{- 1} (\frac{A - B}{1 + A B})$
Given equation $\tan^{- 1} x + \tan^{- 1} y + \tan^{- 1} z = π,$ then
$\begin{matrix} \tan^{- 1} x + \tan^{- 1} y = π - \tan^{- 1} z \\ \tan^{- 1} \frac{x + y}{1 - x y} = (π - \tan^{- 1} z) \\ \frac{x + y}{1 - x y} = \tan (\tan^{- 1} (- z)) \\ \frac{x + y}{1 - x y} = - z \\ x + y = - z (1 - x y) \\ x + y = - z + x y z \\ x + y + z = x y z \end{matrix}$
Therefore, $x + y + z = x y z$ .
Hence, the correct option is (A).

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If tan−1x+tan−1y+tan−1z=π, then x+y+z=

If $\tan^{- 1} x + \tan^{- 1} y + \tan^{- 1} z = π,$ then $x + y + z =$