If x < 0, y < 0 such that xy = 1, then ^ -1 x+ ^ -1 y equals: - Vidyadip

MCQ

If $x < 0, y < 0$ such that $xy = 1,$ then $\tan^{-1}\text{x}+\tan^{-1}\text{y}$ equals:

A
$\frac{\pi}{2}$
✓
$-\frac{\pi}{2}$
C
$-\pi$
D
None of these

✓

Answer

Correct option: B.

$-\frac{\pi}{2}$

We know that $\tan^{-1}\text{x}+\tan^{-1}\text{y}=\tan^{-1}\Big(\frac{\text{x}+\text{y}}{1-\text{xy}}\Big)$
$x < 0, y < 0$ such that
$xy = 1$
Let $x = -a$ and $y = -b,$ where $a$ and $b$ both are positive.
$\tan^{-1}\text{x}+\tan^{-1}\text{y}=\tan^{-1}\Big(\frac{\text{x}+\text{y}}{1-\text{xy}}\Big)$
$=\tan^{-1}\Big(\frac{-\text{a}-\text{a}}{1-1}\Big)$
$=\tan^{-1}(-\infty)$
$=\tan^{-1}\Big\{\tan\Big(-\frac{\pi}{2}\Big)\Big\}$
$=-\frac{\pi}{2}$

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