If ( ^ -1 x )^2+ ( ^ -1 x )^2=5 ^2 8 , then x equals - Vidyadip

MCQ

If $\left(\tan ^{-1} x\right)^2+\left(\cot ^{-1} x\right)^2=\frac{5 \pi^2}{8}$, then $x$ equals

✓
$-1$
B
$1$
C
$0$
D
$4$

✓

Answer

Correct option: A.

$-1$

(A) $\left(\tan ^{-1} x\right)^2+\left(\cot ^{-1} x\right)^2=\frac{5 \pi^2}{8}$
$\Rightarrow\left(\tan ^{-1} x+\cot ^{-1} x\right)^2$ $-2 \tan ^{-1} x\left(\frac{\pi}{2}-\tan ^{-1} x\right)=\frac{5 \pi^2}{8}$
$\Rightarrow \frac{\pi^2}{4}-2 \times \frac{\pi}{2} \tan ^{-1} x+2\left(\tan ^{-1} x\right)^2=\frac{5 \pi^2}{8}$
$\Rightarrow 2\left(\tan ^{-1} x\right)^2-\pi \tan ^{-1} x-\frac{3 \pi^2}{8}=0$
$\Rightarrow \tan ^{-1} x=-\frac{\pi}{4}, \frac{3 \pi}{4}$
$\Rightarrow \tan ^{-1} x=-\frac{\pi}{4} \Rightarrow x=-1$

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