The value of $$\cot \left(\sum_\limits{n=1}^{23} \cot ^{-1}\left(1+\sum_\limits{k=1}^n 2 k\right)\right)$$ is

If $$\cos ^{-1} x-\cos ^{-1} \frac{y}{3}=\alpha$$, where $$-1 \leq x \leq 1, -3 \leq y \leq 3, x \leq \frac{y}{3}$$, then for all $$x, y$$ $$9 x^2-6 x y \cos \alpha+y^2$$ is equal to

If $$\tan ^{-1}\left(\frac{1-x}{1+x}\right)=\frac{1}{2} \tan ^{-1} x$$, then $$x$$ has the value

The value of $$\sin \left(2 \sin ^{-1} 0.8\right)$$ is equal to