Question

If all the roots of the equation $x^{3}-3 x=0$ satisfy the equation $\left(\alpha-\sin ^{-1}(\sin 2)\right) x^{2}-\left(\beta-\tan ^{-1}(\tan 1)\right) x+\gamma^{2}-2 \gamma+1=0$ , then find the value of $|\cot (\beta+\gamma)+\cot \alpha|$.

Answer 1

$\because x=0, \sqrt{3},-\sqrt{3}$ satisfy the $Q.E$.
$\therefore $ It is an identity
$\therefore \alpha-\sin ^{-1}(\sin 2)=0$
$\Rightarrow \alpha=\sin ^{-1} \sin 2=\pi-2 $
$\beta=\tan ^{-1} \tan 1=1 $
$\gamma^{2}-2 \gamma+1=0 \Rightarrow \gamma=1$
$\therefore E=|\cot (\beta+\gamma)+\cot \alpha|$
$=|\cot 2+\cot (\pi-2)|=0$